Hard Level Worksheet Questions

Interactive Square Roots and Cube Roots Worksheet

Square roots and cube roots are fundamental operations in advanced mathematics. Mastering perfect squares, cubes, prime factorization methods, and real-world applications is essential for higher mathematical concepts and problem-solving skills.

Let's start with challenging square root and cube root calculations.

1. Find the value of √(144 × 81).

Step 1: √(144 × 81) = √144 × √81 = 12101514 × 98107

Step 2: Therefore, √(144 × 81) = × = 10896120144

Perfect! Using the property √(a×b) = √a × √b = 12 × 9 = 108.

2. Write the smallest square number that is divisible by 12, 18, and 20.

Step 1: LCM of 12, 18, 20 = 180160200240

Step 2: 180 = 2²23 × 3²3 × 552, to make perfect square multiply by 52310

Step 3: Smallest square number = 180 × 5 = 9008001000950

Excellent! LCM = 180, multiply by 5 to get 900 = 30².

3. Evaluate: √(49/121).

Step 1: √(49/121) = √49/√121 =

Step 2: Therefore, √(49/121) = 7/118/116/115/11

Great! √(a/b) = √a/√b = 7/11.

4. Write the cube root of −512.

Step 1: 512 = 8³84, so ∛512 = 8796

Step 2: Therefore, ∛(-512) = -88-77

Perfect! Cube root of negative number is negative: ∛(-512) = -8.

5. Which smallest number must be added to 930 to make it a perfect square?

Step 1: √930 ≈ , so next perfect square is 31²30²32²29²

Step 2: 31² = , so number to add = 961 - 930 = 31303229

Excellent! Next perfect square after 930 is 961, so add 31.

6. Find the square root of 0.0009.

Step 1: 0.0009 = = 3²/100²9/1003/10009/1000

Step 2: √0.0009 = = 0.030.30.0030.3

Great! √0.0009 = √(9/10000) = 3/100 = 0.03.

7. Which is greater: √90 or 9.5?

Step 1: √90 ≈ 9.499.59.69.4 (since 9.5² = 90.25)

Step 2: Therefore, 9.5√90 is greater

Correct! √90 ≈ 9.49, which is less than 9.5.

8. Find the cube root of 2.197 (correct to 2 decimal places).

Step 1: 2.197 = = 13³/10³12³/10³14³/10³11³/10³

Step 2: ∛2.197 = = 1.301.201.401.10

Perfect! 2197 = 13³, so ∛2.197 = 1.30.

9. Write the next perfect cube after 8000.

Step 1: ∛8000 = 20192118, so next integer is 21202219

Step 2: Next perfect cube = 21³ = 9261900095008800

Excellent! 20³ = 8000, so next cube is 21³ = 9261.

10. If x² = 324, find x.

Step 1: x = ±√324 = ±18171916

Perfect! x = ±18 (both positive and negative values).

Drag each concept to its correct method or property:

Part B: Short Answer Questions (2 Marks Each)

1. Find the least number which must be multiplied to 392 to make it a perfect square.

Step 1: Prime factorization

392 = 2³ × 7²2² × 7³2⁴ × 7¹2¹ × 7⁴

Step 2: Identify unpaired factors

For perfect square, all prime factors need powers. 2³ has power.

Multiply by 24714 to make 2⁴ × 7²

Perfect! 392 × 2 = 784 = 28², so multiply by 2.

2. Find the square root of 5476 by the long division method.

√5476 = 74737576

Excellent! Using long division method gives √5476 = 74.

3. The area of a square field is 1849 m². Find the length of its side.

Step 1: Apply square root

Side length = √1849 = 43 m41 m45 m47 m

Great! √1849 = 43, so side length is 43 meters.

4. Find the cube root of 0.008 using prime factorization method.

Step 1: Express as fraction

0.008 = 8/1000 = 2³/10³2²/10²2⁴/10⁴2¹/10¹

Step 2: Apply cube root

∛0.008 = ∛(2³/10³) = 2/10 = 0.20.022.020

Perfect! ∛0.008 = ∛(8/1000) = 2/10 = 0.2.

5. Which smallest number must be subtracted from 7581 to make it a perfect square?

Step 1: Find nearest perfect square

√7581 ≈ , so nearest perfect square below is 87²86²88²85²

Step 2: Calculate difference

87² = 7569, so subtract: 7581 - 7569 = 1210158

Excellent! Subtract 12 to get 7569 = 87².

Part C: Long Answer Questions (4 Marks Each)

1. The product of two numbers is 1296. If one number is 16, find the other number and check if it's a perfect square.

Step 1: Find the other number

Other number = 1296 ÷ 16 = 81808279

Step 2: Check if it's a perfect square

√81 = 98107

Step 3: Verification

Since 9² = 81, the number isis not a perfect square

Perfect! The other number is 81 = 9², which is a perfect square.

2. A farmer wants to fence a square plot of area 2.25 hectares. Find the side length and cost of fencing at ₹120 per meter.

Step 1: Convert area to m²

2.25 hectares = 2.25 × = 22500 m²225 m²2250 m²225000 m²

Step 2: Find side length

Side length = √22500 = 150 m145 m155 m160 m

Step 3: Calculate perimeter

Perimeter = 4 × = 600 m580 m620 m640 m

Step 4: Calculate cost

Cost = 600 × = ₹72000₹70000₹75000₹68000

Outstanding! Side = 150m, perimeter = 600m, cost = ₹72,000.

3. The sum of squares of two consecutive odd numbers is 290. Find the numbers.

Step 1: Set up equation

Let the numbers be n and n+2 (consecutive odd numbers)

Equation: n² + (n+2)² =

Step 2: Expand and simplify

n² + n²n + + 4 = 290

n² + 4n + 4 = 290

2n² + 4n 286 = 0

Dividing both sides by 2, we get:

n² + - = 0

Step 3: Solve quadratic equation

(n + 13)(n - 11n + 11) = 0

Since n must be positive odd, n = 1113-11-13

Step 4: Find both numbers

First number = 11, Second number = 11 + 2 = 13121415

Step 5: Verify

11² + 13² = 290280300270

Excellent! The consecutive odd numbers are 11 and 13.

4. The surface area of a cube is 600 cm². Find the edge length and volume.

Step 1: Set up surface area formula

Surface area = 6a² =

Therefore, a² = 1009011080

Step 2: Find edge length

Edge length a = √100 = 10 cm9 cm11 cm8 cm

Step 3: Calculate volume

Volume = a³ = 10³ = 1000 cm³900 cm³1100 cm³800 cm³

Perfect! Edge length = 10 cm, Volume = 1000 cm³.

Test your understanding with these multiple choice questions:

For each question, click on the correct answer:

1. The square root of 0.0004 is:

(a) 0.02 (b) 0.2 (c) 2 (d) 20

Correct! √0.0004 = √(4/10000) = 2/100 = 0.02.

2. The cube of 0.5 is:

(a) 0.25 (b) 0.125 (c) 0.05 (d) 0.015

Correct! (0.5)³ = 0.5 × 0.5 × 0.5 = 0.125.

3. The least number to be multiplied to 72 to make it a perfect square is:

(a) 2 (b) 3 (c) 6 (d) 12

Correct! 72 = 2³ × 3². To make perfect square, multiply by 2 to get 2⁴ × 3².

4. The cube root of −2197 is:

(a) -13 (b) 13 (c) -9 (d) 9

Correct! Since 13³ = 2197, we have ∛(-2197) = -13.

5. The smallest perfect cube greater than 100 is:

(a) 121 (b) 125 (c) 216 (d) 343

Correct! 5³ = 125 is the smallest perfect cube greater than 100.

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

• Advanced Square Root Calculations: Product rules, quotient rules, and decimal root

• Complex Cube Root Operations: Negative numbers, decimal expressions, and application

• Prime Factorization Mastery: Finding factors to make perfect squares and cube

• Long Division Method: Step-by-step calculation of square root

• Perfect Square and Cube Analysis: Identifying and creating perfect power

• Real-World Problem Solving: Area, volume, fencing, and arrangement problem

• Estimation Techniques: Finding nearest perfect squares and cube

• Advanced Applications: Consecutive number problems, surface area, and optimization

Your expertise in square roots and cube roots prepares you for advanced algebra, geometry, trigonometry, and calculus concepts!