Square Roots and Cube Roots

Square Roots and Cube Roots

Table of Contents

Square Roots and Cube Roots:

Square root

Imagine a square. The square root of a number is the side length of that square if you were to calculate the area (which is the length times the width). In other words, it’s the number you multiply by itself twice to get the original number.

Cube root

Think of a cube (like a dice). The cube root of a number is the side length of that cube if you were to calculate the volume (which is length times width times height). It’s the number you multiply by itself three times to get the original number.

Why are they important?

These concepts are fundamental building blocks in math. They are used in many areas like:

  • Geometry: calculating areas, volumes, and distances.
  • Algebra: solving equations and understanding proportions.
  • Trigonometry: working with angles and their properties.

Square Root

The square root of a number is like a hidden partner. It’s a number that, when you multiply it by itself (its square), you get the original number back. In other words, if we call the square root b, and the original number a, then b x b = a.

Important to remember: Square roots only apply to non-negative numbers (0 and positive numbers).

Examples

  • The square root of 9 is 3, because 3 x 3 (imagine a square with side lengths of 3) equals 9, the perfect square.
  • The square root of 16 is 4, because 4 x 4 (another square with side lengths of 4) equals 16.

Square Root Notation

We use the symbol √ to represent the square root. So, √9 is the square root of 9, which we know is 3.

Perfect Squares

A perfect square is a number obtained by squaring (multiplying by itself) a whole number. In this case, for the first 30 numbers, the perfect squares are:

  • 1 (√1 = 1)
  • 4 (√4 = 2)
  • 9 (√9 = 3)
  • 16 (√16 = 4)
  • 25 (√25 = 5)

As you can see, the square root of a perfect square is a whole number.

Non-Perfect Squares

The rest of the numbers (2, 3, 5, 6, 7, 8, etc.) are not perfect squares. Their square roots are irrational numbers, meaning they cannot be expressed as a simple fraction and their decimal representation never ends or repeats in a predictable pattern.

  • √2 (approximate decimal value: 1.414)
  • √3 (approximate decimal value: 1.732)
  • √10 (approximate decimal value: 3.162)
Prime Factorization

This method works well for perfect squares with composite numbers (numbers divisible by more than 1 and itself). Here’s how it works:

  • Break down the number into its prime factors (factors that are only divisible by 1 and themselves).
  • Separate the prime factors into pairs.
  • Take the square root of each prime factor within each pair.
  • Multiply the squared factors back together. This will be the square root of the original number.

For example, find the square root of 36:

  • Prime factorize 36: 36 = 2 x 2 x 3 x 3
  • Separate the prime factors into pairs: (2 x 2), (3 x 3)
  • Take the square root of each factor: √(2 x 2) = 2√2, √(3 x 3) = 3√3
  • Multiply the squared factors: 2√2 x 3√3 = 6√12 (simplified form)

Question 1: Find the square root of 10201.

Solution: Let’s use the prime factorization method to find the square root of 10201.

Prime factorization of 10201 = 101×101

10201=101×101

10201=

Question 2: Find the square root of 11664.

Solution: Prime factorization of 11664 = 24×32×72

11664=24×32×72

11664=22×3×7

 

Question 3: Find the square root of 14400.

Solution: Prime factorization of 14400 = 26×32×52

14400=26×32×52

14400=23×3×5 = 120

Question 4: Find the square root of 13689.

Solution: Prime factorization of 13689 = 117×117

13689=117×117

13689= 117

Question 5: Find the square root of 15876.

Solution: Prime factorization of 15876 = 22×32×72×17

15876=22×32×72×17

15876=2×3×7×17=6×7×17=4217

 

Question 6: Find the square root of 22500.

Solution: Prime factorization of 22500 = 22×32×54

22500=22×32×54

22500=2×3×52=30×5 = 150

Question 7: Find the square root of 21025.

Solution: Prime factorization of 21025 = 145×145

21025=145×145

21025=145

Question 8: Find the square root of 12321.

Solution: Prime factorization of 12321 = 111×111

12321=111×111

12321

Question 9: Find the square root of 17424.

Solution: Prime factorization of 17424 = 24×32×112

17424=24×32×112

17424=22×3×11=

Question 10: Find the square root of 21025.

Solution: Prime factorization of 21025 =145×145

21025=145×145

21025=

Cube Roots

Think of a cube: six squares forming a perfect three-dimensional shape. Cube roots are like finding the secret side length that, when used to build the cube, results in the original volume. In other words, a cube root is the inverse operation of cubing a number.

Finding the Root of the Cube

The cube root of a number is a value that, when multiplied by itself three times (cubed), equals the original number. We can call this value b, and the original number a. So, b x b x b = a. Important to remember: Cube roots apply to all numbers, positive, negative, and zero.

Examples

  • The cube root of 27 is 3, because 3 x 3 x 3 (imagine a cube with sides of length 3) equals 27, a perfect cube.
  • The cube root of -64 is -4, because -4 x -4 x -4 equals -64, another perfect cube (though negative).
  • The cube root of 8 is 2, since 2 x 2 x 2 equals 8.

Cube Root Notation

We use the symbol ³√ to represent the cube root. So, ³√27 is the cube root of 27, which we know is 3.

Perfect Cubes

This is the simplest method, similar to perfect squares. If you’re dealing with a perfect cube (a number obtained by cubing an integer), you can simply recognize its cube root. For example, the cube root of 27 is 3 because 3 x 3 x 3 = 27. You can memorize the perfect cubes from 1 to a certain number for quick reference.

Prime Factorization

This method works well for perfect cubes with composite numbers (numbers divisible by more than 1 and itself). Here’s the breakdown:

For example, find the cube root of 64:

  • Prime factorize 64: 64 = 2 x 2 x 2 x 2 x 2 x 2
  • Separate the prime factors into groups of three: (2 x 2 x 2), (2 x 2 x 2)
  • Take the cube root of each factor: √(2 x 2 x 2) = 2√2, √(2 x 2 x 2) = 2√2 (because each 2 is cubed)
  • Multiply the cubed factors: 2√2 x 2√2 = 4√4 (simplified form)

Question 1: Find the cube root of 32768.

Solution and Steps: 327683

  • Prime factors of 32768=2×2×2×2×2×2×2×2×2×2×2×2
  • Group the prime factors in triples.
  • 327683=2×2×23×2×2×23×2×2×23×2×2×23×2×2×23×2×2×23
  • 327683=2×2×2=

Answer: The cube root of 32768 is 8.

Question 2: Find the cube root of 17576.

Solution and Steps: 175763

  • Identify the prime factors of 17576=2×2×2×7×7×7
  • Group the prime factors in triples.
  • 175763=2×2×23×7×7×73×7×7×73
  • 175763=2×7×7=

Answer: The cube root of 17576 is 98.

Question 3: Find the cube root of 27000.

Solution and Steps: 270003

  • Identify the prime factors of 27000=2×2×2×3×3×3×5×5×5
  • Group the prime factors in triples.
  • 270003=2×2×23×3×3×33×5×5×53×2×3×53
  • 270003=2×3×5=

Answer: The cube root of 27000 is 30.

Question 4: Find the cube root of 42875.

Solution and Steps: 428753

  • Identify the prime factors of 42875=5×5×5×5×13
  • Group the prime factors in triples.
  • 428753=5×5×53×5×5×133
  • 428753=5×5×133
  • 428753=5×3.819.

Answer: The cube root of 42875 is approximately 19.3

Question 5: Find the cube root of 74088.

Solution and Steps 740883

  • Identify the prime factors of 74088 = 2×2×2×3×19×163
  • Group the prime factors in triples.
  • 740883=2×2×23×3×19×1633
  • 740883=2×3×19×1633
  • 740883=2×11.923.8

Answer: The cube root of 74088 is approximately 23.8

Question 6: Find the cube root of 97336.

Solution and Steps 973363

  • Identify the prime factors of 97336=2×2×2×13×37×41
  • Group the prime factors in triples.
  • 973363=2×2×23×13×37×413
  • 973363=2×13×37×413
  • 973363=2×28.957.8

Answer: The cube root of 97336 is approximately 57.8

Question 7: Find the cube root of 106120.

Solution and Steps 1061203

  • Identify the prime factors of 106120=2×2×2×5×53×199
  • Group the prime factors in triples.
  • 1061203=2×2×23×5×53×1993
  • 1061203=2×5×53×1993
  • 1061203=2×34.669.2

Answer: The cube root of 106120 is approximately 69.2

Question 8: Find the cube root of 121670.

Solution and Steps 1216703

  • Identify the prime factors of 121670=2×5×7×17×31
  • Group the prime factors in triples.
  • 1216703=2×5×73×17×313
  • 1216703=2×5×73×17×313
  • 1216703=2×3.9×15.5120.9

Answer: The cube root of 121670 is approximately 120.9

Question 9: Find the cube root of 148877.

Solution and Steps: 1488773

  • Identify the prime factors of 148877=11×37×367
  • Group the prime factors in triples.
  • 1488773=11×37×3673
  • 1488773=11×373×3673
  • 1488773=3.3×7.625.1

Answer: The cube root of 148877 is approximately 25.1

Question 10: Find the cube root of 175616.

Solution and Steps: 1756163

  • Identify the prime factors of 175616=2×2×2×2×2×11×11×11
  • Group the prime factors in triples.
  • 1756163=2×2×23×2×2×23×11×11×113
  • 1756163=2×2×11=

Answer: The cube root of 175616 is 44.

MCQ’s

  1. What is the square root of 25?

    • (a) 3
    • (b) 5 ✅
    • (c) 7
    • (d) 10
  2. Which of the following is NOT a perfect square?

    • (a) 49
    • (b) 64
    • (c) 81 ✅
    • (d) 100
  3. The cube root of 8 is:

    • (a) 1
    • (b) 2 ✅
    • (c) 3
    • (d) 4
  4. What is the notation used to represent the square root?

    • (a) ³√
    • (b) √ ✅
    • (c) ^
    • (d) %
  5. The square root of a negative number is:

    • (a) Always a real number
    • (b) Always an imaginary number (not real) ✅
    • (c) Sometimes real, sometimes imaginary
    • (d) None of the above
  6. What is the approximate value of √20 (rounded to one decimal place)?

    • (a) 3.5
    • (b) 4.2 ✅
    • (c) 4.8
    • (d) 5.1
  7. Which of the following expressions represents the cube root of 125?

    • (a) 125√
    • (b) ³√125 ✅
    • (c) (125)^(1/3)✅
    • (d) All of the above
  8. The product of two perfect squares is always a:

    • (a) Prime number
    • (b) Composite number
    • (c) Perfect cube
    • (d) Perfect square ✅
  9. A cube root of 64 is:

    • (a) 2 ✅
    • (b) -2
    • (c) 4
    • (d) None of the above
  10. Which of the following is the smallest perfect square greater than 40?

    • (a) 25
    • (b) 36
    • (c) 49 ✅
    • (d) 64

Answer Key:

  1. (b)
  2. (c)
  3. (b)
  4. (b)
  5. (b)
  6. (b)
  7. (b) & (c)
  8. (d)
  9. (a)
  10. (c)
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