Cube Roots: Definition, Examples, and Calculation

Lesson Overview

• What Are Cube Roots?
• Cube Root Formula
• Positive and Negative Cube Roots
• How to Calculate the Cube Root of a Number
• Differences Between Square Roots and Cube Roots
• Solved Examples On Cube Roots

Cube roots are the reverse of cubing a number. When you cube a number, you multiply it by itself three times. For example, 3 cubed (3^3) equals 27. The cube root of 27 is the number that, when cubed, gives 27 - in this case, 3.

Cube roots are useful for solving problems involving volumes and exponents.  It determines the side length of a cube given its volume.

What Are Cube Roots?

A cube root is a number that, when multiplied by itself three times, yields the original number. More formally, the cube root of a number 'x' is the number 'y' that satisfies the equation y³ = x. The cube root is denoted by the symbol ³√x.

Example:

• ³√216 = 6, since 6 x 6 x 6 = 216

The concept of cube roots extends to all real numbers, including negative numbers. It is essential in various mathematical applications, including solving volume equations and understanding higher-order polynomials.

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Cube Root Formula

The cube root formula is used to determine the cube root of any number, often represented in radical form with the symbol ∛. 

To calculate it, the number is first broken down into its prime factors, and the cube root formula is applied. 

If x is a number such that x = y × y × y, then:

Cube root of x = ∛x = ∛(y × y × y) = y

Here, y represents the cube root of x. If y is an integer, then x is classified as a perfect cube.

Applications of the Cube Root Formula

• Solving cubic equations.
• Determining the side length of a cube when the volume is provided.

Positive and Negative Cube Roots

Cube roots can be both positive and negative, depending on the sign of the original number. 

Positive Cube Roots

When a positive number is cubed, the result is positive. Therefore, the cube root of a positive number is also positive.  

For example: Find the cube root of 8.

1. Write the cube root expression: ∛8.
2. Identify the number that multiplies by itself three times to give 8:
   2 × 2 × 2 = 8.
3. Therefore, the cube root of 8 is 2.

Negative Cube Roots

When a negative number is cubed, the result is negative. This means the cube root of a negative number is also negative. 

For example: Find the cube root of -27.

1. Write the cube root expression: ∛(-27).
2. Identify the number that multiplies by itself three times to give -27:
   -3 × -3 × -3 = -27.
3. Therefore, the cube root of -27 is -3.

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Squares And Cubes Quiz

How to Calculate the Cube Root of a Number

The cube root of a number can be calculated using the prime factorization method.

Step 1: Perform the prime factorization of the given number.
Step 2: Group the prime factors into sets of three identical factors.
Step 3: For each group of three, take one factor out of the cube root symbol. Multiply these factors to get the cube root. If a factor cannot form a group of three, the number is not a perfect cube.

Example: Find the cube root of 729.

1. Prime factorization of 729: 3 × 3 × 3 × 3 × 3 × 3.
2. Group the factors: (3 × 3 × 3) and (3 × 3 × 3).
3. Take one factor from each group: 3 × 3 = 9.

So, the cube root of 729 is 9.

Differences Between Square Roots and Cube Roots

Square roots and cube roots are mathematical operations with distinct purposes:

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Challenging quiz on Square and Cube Roots

Solved Examples On Cube Roots

Example 1: Find the cube root of 64.

Solution:

1. Perform prime factorization of 64:
   64 = 2 × 2 × 2 × 2 × 2 × 2.
2. Group the factors into three identical sets: (2 × 2 × 2) and (2 × 2 × 2).
3. Take one factor from each group: 2 × 2 = 4.
   Answer: The cube root of 64 is 4.

Example 2: Find the cube root of 125.

Solution:

1. Prime factorization of 125:
   125 = 5 × 5 × 5.
2. Group the factors into one set of three identical factors: (5 × 5 × 5).
3. Take one factor out: 5.
   Answer: The cube root of 125 is 5.

Example 3: Find the cube root of -216.

Solution:

1. Prime factorization of 216:
   216 = 2 × 2 × 2 × 3 × 3 × 3.
2. Add the negative sign, as the number is negative.
3. Group the factors into identical sets: (2 × 2 × 2) and (3 × 3 × 3).
4. Take one factor from each group: -2 × 3 = -6.
   Answer: The cube root of -216 is -6.

Example 4: Find the cube root of 1.

Solution:

1. Prime factorization of 1:
   1 = 1 × 1 × 1.
2. Group the factors: (1 × 1 × 1).
3. Take one factor out: 1.
   Answer: The cube root of 1 is 1.

Example 5: Determine if 1000 is a perfect cube and find its cube root if it is.

Solution:

1. Prime factorization of 1000:
   1000 = 2 × 2 × 2 × 5 × 5 × 5.
2. Group the factors into identical sets: (2 × 2 × 2) and (5 × 5 × 5).
3. Take one factor from each group: 2 × 5 = 10.

Since all factors were grouped perfectly, 1000 is a perfect cube.
Answer: The cube root of 1000 is 10.