{"id":1952,"date":"2026-05-23T10:34:46","date_gmt":"2026-05-23T10:34:46","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=1952"},"modified":"2026-05-23T10:34:50","modified_gmt":"2026-05-23T10:34:50","slug":"ncert-class-8-a-square-and-a-cube-solutions-guide","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/ncert-class-8-a-square-and-a-cube-solutions-guide\/","title":{"rendered":"NCERT Class 8 A Square and A Cube Solutions Guide"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Short Introduction<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">This chapter explains important mathematical concepts like Perfect Squares, Square Roots, Perfect Cubes, and Cube Roots in a simple and logical way. Below are detailed step-by-step solutions for all important questions from the chapter \u201cA Square and A Cube\u201d in a portal-ready SEO format.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Quick Information Box<\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Particular<\/th><th>Details<\/th><\/tr><\/thead><tbody><tr><td>Chapter Name<\/td><td>A Square and A Cube<\/td><\/tr><tr><td>Class<\/td><td>Grade 8<\/td><\/tr><tr><td>Subject<\/td><td>Mathematics<\/td><\/tr><tr><td>Main Topics<\/td><td>Perfect Squares, Cubes, Square Roots, Cube Roots<\/td><\/tr><tr><td>Difficulty Level<\/td><td>Easy to Moderate<\/td><\/tr><tr><td>Important For<\/td><td>School Exams, Olympiads, Scholarships<\/td><\/tr><tr><td>Learning Outcome<\/td><td>Understanding number patterns and roots<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Concepts Used (Topics Covered)<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Perfect Squares<\/li>\n\n\n\n<li>Perfect Cubes<\/li>\n\n\n\n<li>Square Roots<\/li>\n\n\n\n<li>Cube Roots<\/li>\n\n\n\n<li>Prime Factorisation<\/li>\n\n\n\n<li>Properties of Squares<\/li>\n\n\n\n<li>Properties of Cubes<\/li>\n\n\n\n<li>Odd Number Patterns<\/li>\n\n\n\n<li>Estimation of Roots<\/li>\n\n\n\n<li>Number Patterns<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Important Formulas<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Square Formula<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mi>n<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n^2=n\\times n<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Cube Formula<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>n<\/mi><mn>3<\/mn><\/msup><mo>=<\/mo><mi>n<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n^3=n\\times n\\times n<\/annotation><\/semantics><\/math>n3=n\u00d7n\u00d7n<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Square Root<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><msup><mi>n<\/mi><mn>2<\/mn><\/msup><\/msqrt><mo>=<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{n^2}=n<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Cube Root<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><msup><mi>n<\/mi><mn>3<\/mn><\/msup><mn>3<\/mn><\/mroot><mo>=<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{n^3}=n<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Sum of Consecutive Odd Numbers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mo>+<\/mo><mn>5<\/mn><mo>+<\/mo><mo>\u22ef<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">1+3+5+\\cdots +(2n-1)=n^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Questions &amp; Step-by-Step Solutions<\/h1>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 2<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q1. Does every number have an even number of factors?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">No. Perfect square numbers have an odd number of factors.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Example:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Factors of 4 \u2192 1, 2, 4<\/li>\n\n\n\n<li>Total factors = 3 (odd)<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">No, every number does not have an even number of factors.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q2. Can you use this insight to find more numbers with an odd number of factors?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Yes. All perfect square numbers have an odd number of factors.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Examples:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>1<\/li>\n\n\n\n<li>4<\/li>\n\n\n\n<li>9<\/li>\n\n\n\n<li>16<\/li>\n\n\n\n<li>25<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">All square numbers have an odd number of factors.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 3<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q3. Write the locker numbers that remain open.<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Only lockers with perfect square numbers remain open because they are toggled an odd number of times.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The perfect squares up to 100 are:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mo separator=\"true\">,<\/mo><mn>4<\/mn><mo separator=\"true\">,<\/mo><mn>9<\/mn><mo separator=\"true\">,<\/mo><mn>16<\/mn><mo separator=\"true\">,<\/mo><mn>25<\/mn><mo separator=\"true\">,<\/mo><mn>36<\/mn><mo separator=\"true\">,<\/mo><mn>49<\/mn><mo separator=\"true\">,<\/mo><mn>64<\/mn><mo separator=\"true\">,<\/mo><mn>81<\/mn><mo separator=\"true\">,<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1,4,9,16,25,36,49,64,81,100<\/annotation><\/semantics><\/math>1,4,9,16,25,36,49,64,81,100<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">1, 4, 9, 16, 25, 36, 49, 64, 81, 100<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 5<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q4. Which of the following numbers have the digit 6 in the units place?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>38\u00b2<\/li>\n\n\n\n<li>34\u00b2<\/li>\n\n\n\n<li>46\u00b2<\/li>\n\n\n\n<li>56\u00b2<\/li>\n\n\n\n<li>74\u00b2<\/li>\n\n\n\n<li>82\u00b2<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A number ending in 4 or 6 gives a square ending in 6.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>34\u00b2 \u2714<\/li>\n\n\n\n<li>46\u00b2 \u2714<\/li>\n\n\n\n<li>56\u00b2 \u2714<\/li>\n\n\n\n<li>74\u00b2 \u2714<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">34\u00b2, 46\u00b2, 56\u00b2, 74\u00b2<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q5. If a number contains 3 zeros at the end, how many zeros will its square have?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Example:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>1000<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>1000000<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1000^2=1000000<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The square has 6 zeros.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Six zeros.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q6. What do you notice about zeros at the end of a number and its square?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The number of zeros at the end of a square is double the zeros at the end of the original number.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Squares always have an even number of zeros at the end.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q7. What can you say about the parity of a number and its square?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Square of an even number is even.<\/li>\n\n\n\n<li>Square of an odd number is odd.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Square of an even number is even and square of an odd number is odd.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 7<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q8. Find how many numbers lie between two consecutive perfect squares.<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For consecutive perfect squares:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mtext>&nbsp;and&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">n^2\\text{ and }(n+1)^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Numbers between them:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>2<\/mn><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(n+1)^2-n^2-1=2n<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">There are 2n numbers between consecutive perfect squares.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Figure It Out (Page 10)<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q1. Which of the following numbers are not perfect squares?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Numbers:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2032, 2048, 1027, 1089<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">1089 = 33\u00b2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The remaining numbers are not perfect squares.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">2032, 2048, 1027<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q2. Which one among 64\u00b2, 108\u00b2, 292\u00b2, 36\u00b2 has last digit 4?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Numbers ending in 2 or 8 have squares ending in 4.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">108\u00b2 and 292\u00b2<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q3. Given 125\u00b2 = 15625, find 126\u00b2<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using identity:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>a<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(a+1)^2=a^2+2a+1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">126\u00b2 = 125\u00b2 + 2\u00d7125 +1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">= 15625 + 251<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">= 15876<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">15876<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q4. Find the side of a square whose area is 441 m\u00b2<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mn>441<\/mn><\/msqrt><mo>=<\/mo><mn>21<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{441}=21<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">21 m<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q5. Find the smallest square number divisible by 4, 9 and 10<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">LCM of 4, 9 and 10:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">= 180<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Prime factors:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">180 = 2\u00b2 \u00d7 3\u00b2 \u00d7 5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To make it a perfect square, multiply by another 5.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">180 \u00d7 5 = 900<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">900<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q6. Find the smallest number by which 9408 must be multiplied to make a perfect square<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Prime factorisation:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">9408 = 2\u2076 \u00d7 3 \u00d7 7\u00b2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">3 has no pair.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Multiply by 3.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">9408 \u00d7 3 = 28224<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Square root:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mn>28224<\/mn><\/msqrt><mo>=<\/mo><mn>168<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{28224}=168<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Multiplier = 3<br>Square Root = 168<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q7. How many numbers lie between:<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">(i) 16\u00b2 and 17\u00b2<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using formula:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2 \u00d7 16 = 32<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">32<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(ii) 99\u00b2 and 100\u00b2<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">2 \u00d7 99 = 198<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">198<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q8. Fill in the missing numbers<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>4<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>5<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>20<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mn>21<\/mn><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">4^2+5^2+20^2=21^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>9<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>10<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>90<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mn>91<\/mn><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">9^2+10^2+90^2=91^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answers<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">21, 90, 91<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 13<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q9. Can a cube end with exactly two zeroes?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">No. Cubes always contain zeros in multiples of 3.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">No<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 14<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q10. Find the sum:<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">91 + 93 + 95 + &#8230; + 109<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">This pattern represents cube numbers.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>10<\/mn><mn>3<\/mn><\/msup><mo>=<\/mo><mn>1000<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10^3=1000<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 15<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q11. Find cube roots<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">(i)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>64<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{64}=4<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(ii)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>512<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{512}=8<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(iii)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>729<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{729}=9<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 16<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q12. Find the cube roots of 27000 and 10648<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>27000<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>30<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{27000}=30<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>10648<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>22<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{10648}=22<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q13. What number will you multiply by 1323 to make it a cube number?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">1323 = 3\u00b3 \u00d7 7\u00b2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One more 7 is required.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">7<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q14. State True or False<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Statement<\/th><th>Answer<\/th><\/tr><\/thead><tbody><tr><td>Cube of any odd number is even<\/td><td>False<\/td><\/tr><tr><td>No perfect cube ends with 8<\/td><td>False<\/td><\/tr><tr><td>Cube of a 2-digit number may be a 3-digit number<\/td><td>False<\/td><\/tr><tr><td>Cube of a 2-digit number may have seven or more digits<\/td><td>False<\/td><\/tr><tr><td>Cube numbers have odd number of factors<\/td><td>False<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q15. Guess the cube roots<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Answers<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>1331<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{1331}=11<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>4913<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>17<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{4913}=17<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>12167<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>23<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{12167}=23<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mroot><mn>32768<\/mn><mn>3<\/mn><\/mroot><mo>=<\/mo><mn>32<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{32768}=32<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Page No. 17<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q16. Which of the following is greatest?<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Cube differences increase rapidly.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>67<\/mn><mn>3<\/mn><\/msup><mo>\u2212<\/mo><msup><mn>66<\/mn><mn>3<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">67^3-66^3<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is the greatest.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">67\u00b3 \u2212 66\u00b3<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Common Mistakes<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Confusing square numbers with cube numbers<\/li>\n\n\n\n<li>Incorrect prime factorisation<\/li>\n\n\n\n<li>Forgetting pairing and triplet grouping<\/li>\n\n\n\n<li>Wrong estimation of roots<\/li>\n\n\n\n<li>Ignoring units digit rules<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Exam Tips<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Memorise squares up to 30<\/li>\n\n\n\n<li>Memorise cubes up to 20<\/li>\n\n\n\n<li>Practice prime factorisation daily<\/li>\n\n\n\n<li>Learn units digit shortcuts<\/li>\n\n\n\n<li>Revise odd number patterns carefully<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Practice MCQs<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">1. Which is a perfect square?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A. 48<br>B. 64<br>C. 72<br>D. 98<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">B. 64<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">2. Cube root of 125 is:<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A. 4<br>B. 5<br>C. 6<br>D. 7<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">B. 5<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">3. Which number is not a perfect cube?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A. 27<br>B. 64<br>C. 81<br>D. 125<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">C. 81<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">FAQ Section<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Q1. What is a perfect square?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A number obtained by multiplying a number by itself.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q2. What is a perfect cube?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A number obtained by multiplying a number by itself three times.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q3. Can a perfect square end with 7?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">No.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q4. Can a cube end with 8?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Yes.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Q5. What is the square root of 169?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mn>169<\/mn><\/msqrt><mo>=<\/mo><mn>13<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{169}=13<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\udcd8 Get more NCERT Solutions, Mock Tests, Practice Questions, and Exam Preparation Resources at:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><a>www.mymockmate.com<\/a><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\ude80 Improve your preparation with All India Mock Tests and Instant Results.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n    <div class=\"xs_social_share_widget xs_share_url after_content \t\tmain_content  wslu-style-1 wslu-share-box-shaped wslu-fill-colored wslu-none wslu-share-horizontal wslu-theme-font-no wslu-main_content\">\n\n\t\t\n        <ul>\n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"<p>Short Introduction This chapter explains important mathematical concepts like Perfect Squares, Square Roots, Perfect Cubes, and Cube Roots<\/p>\n","protected":false},"author":1,"featured_media":1953,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_angie_page":false,"_surecart_dashboard_logo_width":"180px","_surecart_dashboard_show_logo":true,"_surecart_dashboard_navigation_orders":true,"_surecart_dashboard_navigation_invoices":true,"_surecart_dashboard_navigation_subscriptions":true,"_surecart_dashboard_navigation_downloads":true,"_surecart_dashboard_navigation_billing":true,"_surecart_dashboard_navigation_account":true,"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"page_builder":"","footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[120,119,105],"tags":[452,476,472,458,457,451,461,456,474,465,470,473,459,469,464,475,481,478,466,47,477,480,468,454,467,453,462,479,471,460,455,463],"class_list":["post-1952","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-a-square-and-a-cube","category-class-8","category-maths-class-8","tag-a-square-and-a-cube","tag-algebra-basics","tag-cbse-class-8","tag-cbse-maths-solutions","tag-class-8-mathematics","tag-class-8-maths","tag-cube-numbers","tag-cube-root","tag-educational-blog","tag-exam-preparation","tag-grade-8-maths","tag-math-tricks","tag-maths-chapter-1","tag-maths-formulas","tag-maths-mcqs","tag-maths-practice-questions","tag-maths-tutorial","tag-maths-worksheet","tag-ncert-class-8-maths","tag-ncert-solutions","tag-number-system","tag-online-maths-learning","tag-perfect-cube-questions","tag-perfect-cubes","tag-perfect-square-questions","tag-perfect-squares","tag-prime-factorisation","tag-school-mathematics","tag-square-and-cube-solutions","tag-square-numbers","tag-square-root","tag-step-by-step-solutions"],"jetpack_publicize_connections":[],"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1952","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=1952"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1952\/revisions"}],"predecessor-version":[{"id":1954,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1952\/revisions\/1954"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/1953"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=1952"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=1952"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=1952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}