Moderate Level Worksheet Questions

Square roots and cube roots are fundamental operations in mathematics. Understanding their properties and calculation methods is essential for advanced mathematical concepts.

First, let's explore basic square roots, cube roots, and their properties.

1. Write the square root of 144. 12141610

Awesome! √144 = 12 because 122 = 144.

2. Which of the following is a perfect square: 50, 64, 75? 645075None

Great job! 64 = 82 is a perfect square.

3. Write the cube of 7. 3434921729

Perfect! 73 = 7 × 7 × 7 = 343.

4. Find the square of 1.5. 2.251.253.02.5

Excellent! 1.52 = 1.5 × 1.5 = 2.25.

5. Express 81 as the square of a number. 928272102

Super! 81 = 92 because 9 × 9 = 81.

6. Write the smallest 3-digit perfect cube. 125100108216

That's correct! 125 = 53 is the smallest 3-digit perfect cube.

7. What is the cube root of 512? 8679

Well done! ∛512 = 8 because 83 = 512.

8. Write 0.25 as a square of a number in decimal form. 0.5²0.25²0.420.62

Brilliant! 0.25 = 0.52 because 0.5 × 0.5 = 0.25.

9. Find the square root of 225. 15141613

You nailed it! √225 = 15 because 152 = 225.

10. Find the cube root of –27. -33-99

Perfect! ∛(-27) = -3 because −33 = -27.

Drag each rule to its correct category:

Part B: Short Answer Questions (2 Marks Each)

1. Find the square root of 4624 by the prime factorization method.

Step 1: Prime factorization

4624 = 2 ×

= 22 ×

= 24 ×

= 24 × 17217

Step 2: Apply square root

√4624 = √(2⁴ × 172) = 222 × 17172 =

Excellent! √4624 = 68 using prime factorization.

2. Find the least number that must be multiplied with 2028 to make it a perfect square.

Step 1: Prime factorization

2028 = 4 × = 4 × 3 × = 22 × 3 × 13213

Step 2: Identify unpaired factors

2028 = 22 × 31 × 132

The factor appears with odd power

Step 3: Find missing factor

To make perfect square, multiply by

Check: 2028 × 3 = 6084 = 782

Perfect! The least number is 3.

3. Evaluate √(1/49) and write your answer in simplest form.

Step 1: Apply square root to fraction

√(149) = /

Step 2: Calculate individual roots

√1 = and √49 =

Step 3: Simplify

√(149) =

Great work! √(149) = 1/7.

4. Find the cube root of 3375 by the prime factorization method.

Step 1: Prime factorization

3375 = 3 × = 32 × = 33 ×

= 33 × 5352

Step 2: Apply cube root

∛3375 = ∛(33 × 53) = × =

Outstanding! ∛3375 = 15.

5. Without finding the cube, find the cube root of 43 × 53 × 63.

Step 1: Apply cube root property

∛(43 × 53 × 63) = ∛(43) × ∛(53) × ∛(63)

Step 2: Simplify individual cube roots

∛(43) = , ∛(53) = , ∛(63) =

Step 3: Multiply results

∛(43 × 53 × 63) = 4 × 5 × 6 =

Excellent! The answer is 120.

Part C: Long Answer Questions (4 Marks Each)

1. By long division method, find the square root of 7225.

Step 1: Group digits from right

7225 → |

Step 2: Find first digit

For 72: 82 = 64 < 72 < 81 = 92

So first digit =

Step 3: Calculate remainder

72 - 64 =

Bring down 25:

Step 4: Find second digit

Double first digit: 2 × 8 =

Trial: (160 + x) × x = 825

Try x = 5: (160 + 5) × 5 = 165 × 5 =

Step 5: Final answer

√7225 =

Perfect! √7225 = 85 by long division method.

2. Find the least number which must be added to 1650 to make it a perfect square and find the square root of the resulting number.

Step 1: Find nearest perfect square

√1650 ≈

Check: 402 = < 1650

Check: 412 = > 1650

Step 2: Calculate difference

Number to add = 1681 - 1650 =

Step 3: Find square root

Resulting number = 1650 + 31 =

√1681 =

Great work! Add 31 to get 1681, and √1681 = 41.

3. By prime factorization, find the cube root of 74088.

Step 1: Prime factorization

74088 = 8 × = 8 × 9 ×

= 8 × 9 × 3 × = 8 × 27 ×

= × ×

Step 2: Apply cube root

∛74088 = ∛(2³ × 3³ × 7³) = × × =

Excellent! ∛74088 = 42.

4. A farmer has a square plot of land whose area is 1369 m². He wants to fence it completely. Find the length of the side and the total length of fencing required.

Step 1: Find side length

Area = side2

Side length = √1369 = m

Step 2: Calculate perimeter

Perimeter = 4 × side length

Perimeter = 4 × 37 = m

Step 3: Answer

Side length = m

Total fencing required = m

Perfect! Side = 37 m, fencing = 148 m.

5. The volume of a cube is 13824 cm³. Find the length of each edge of the cube.

Step 1: Set up equation

Volume = edge3

Edge length = ∛13824

Step 2: Prime factorization

13824 = 26 × = 26 × 63 = 2⁶ × 2×33

= 26 × 23 × 33 = 29218 × 33 = 233 × 33 = 8382 × 33

Step 3: Apply cube root

∛13824 = ∛(83 × 33) = × = cm

Outstanding! Each edge is 24 cm long.

Test your understanding with these multiple choice questions:

For each question, click on the correct answer:

1. The square root of 169 is:

(a) 12 (b) 14 (c) 13 (d) 15

Correct! √169 = 13 because 132 = 169.

2. The cube root of 216 is:

(a) 5 (b) 6 (c) 8 (d) 7

Correct! ∛216 = 6 because 63 = 216.

3. Which of the following numbers is a perfect cube?

(a) 125 (b) 128 (c) 150 (d) 100

Correct! 125 = 53 is a perfect cube.

4. The square root of 0.81 is:

(a) 0.8 (b) 0.81 (c) 0.9 (d) 9

Correct! √0.81 = 0.9 because 0.92 = 0.81.

5. The least number to be multiplied with 90 to make it a perfect square is:

(a) 2 (b) 3 (c) 5 (d) 10

Correct! 90 = 2 × 32 × 5, so we need to multiply by 10 (= 2 × 5) to get 900 = 302.

🎉 Outstanding! You've Mastered Square Roots and Cube Roots! Here's what you accomplished:

Understanding perfect squares and perfect cubes Mastering prime factorization methods for finding roots Applying long division method for square roots Solving real-world problems involving areas and volumes Working with decimal square roots and cube roots Understanding properties of square and cube root operations Finding least numbers to make perfect squares and cubes

Your solid foundation in roots will help you excel in algebra, geometry, and advanced mathematical concepts!