Cube Root - Math Steps, Examples & Questions

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Cube root

Here is everything you need to know about cube roots. You will learn what a cube root is, what a cube number is, and how to simplify expressions using cube roots.

Students first work with cube roots in $8$ th grade, when they work with square roots and expand this knowledge as they progress through high school.

What is a cube number and a cube root?

A cube number is a number or variable that is ‘cubed’ which means it is multiplied by itself three times.

The cube root of a number is the value being multiplied by itself three times. Cube rooting a number is the inverse operation of cubing a number.

The cube root function looks like this $\sqrt[3]{\quad}$ where $3$ is the index of the root. The cube root sign can also be called a radical sign.

The cube root can also be expressed with the exponent, $\cfrac{1}{3} \, .$

For example, $4 \times 4 \times 4=64$

This equation can be rewritten to be $4^3=64.$

$4$ is the cube root, and $64$ is the perfect cube number. This is because $4$ is the number being cubed or multiplied by itself three times, and $64$ is the product, so it is the perfect cube.

Therefore, the cube root of $64$ is $4$ which can be written as

$\sqrt[3]{64}=4$ or $(64)^{\frac{1}{3}}=4$

The table below has some of the perfect cube numbers and cube roots.

So, anything multiplied by itself three times forms a perfect cube.

x \times x \times x=x^3

$x^3$ is a perfect cubed algebraic expression where $x$ is the cube root.

It can be useful to see how a cube number relates to an actual cube.

Let’s look at:

\begin{aligned} & 2 \times 2 \times 2=2^3 \\\\ & 3 \times 3 \times 3=3^3 \\\\ & 4 \times 4 \times 4=4^3 \end{aligned}

Each cubed relationship can be represented as an array which forms the shape of a cube that has a length $2$ units, width $2$ units, and depth $2$ units, etc.

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Cubing negative numbers

Any number can be cubed including decimals, fractions and integers.

For example:

$\cfrac{1}{3} \, \times \cfrac{1}{3} \, \times \cfrac{1}{3} \, =\cfrac{1}{27} \, \rightarrow \cfrac{1}{27} \,$ is a perfect cube.

$(-5) \times(-5) \times(-5)=(-125) \rightarrow-125$ is a perfect cube.

(0.2) \times(0.2) \times(0.2)=0.008

You will notice that when you cube a negative number, you get a negative number, and when you cube a positive number, you get a positive number.

This is because a negative number multiplied by a negative number multiplied by a negative number yields a negative result and a positive number multiplied by a positive number, multiplied by a positive number yields a positive result.

Likewise, the cube root of a positive number is a positive root and the cube root of a negative number is a negative root.

For example, $\sqrt[3]{-125}=-5$ (radical form) OR $(-125)^{\frac{1}{3}}=-5$ (exponential form)

What is a cube number and a cube root?

Common Core State Standards

How does this relate to $8$ th grade math and high school math?

Grade 8 Expressions and Equations (8.EE.A.1)
Know and apply the properties of integer exponents to generate equivalent numerical expressions.
For example, $32 \times 3-5 = 3-3 = \cfrac{1}{33} = \cfrac{1}{27}$

Grade 8 Expressions and Equations (8.EE.A.1)
Use square root and cube root symbols to represent solutions to equations of the form $x2 = p$ and $x3 = p,$ where $p$ is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that $\sqrt{2}$ is irrational.

High School Number and Quantity: The Real Number System (HSN-RN.A.1)
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

For example, we define $\cfrac{51}{3}$ to be the cube root of $5$ because we want $(\cfrac{51}{3})^3 = 5(\cfrac{1}{3})^3 \,$ to hold, so $(\cfrac{51}{3})^3$ must equal $5.$

How to cube expressions

In order to cube expressions:

Take the expression and multiply it by itself three times.
Write the cubed expression.

Cube root examples

Example 1: cube positive number

Cube the number $12.$

Take the expression and multiply it by itself three times.

12 \times 12 \times 12=1,728

2Write the cubed expression.

12^3=1,728

Example 2: cube a negative fraction

Cube the number $\left(-\cfrac{2}{3}\right).$

Take the expression and multiply it by itself three times.

\left(-\cfrac{2}{3}\right) \times\left(-\cfrac{2}{3}\right) \times\left(-\cfrac{2}{3}\right)=\left(-\cfrac{8}{27}\right)

Write the cubed expression.

\left(-\cfrac{2}{3}\right)^3=\left(-\cfrac{8}{27}\right)

Example 3: cube an expression

Cube the expression $2x^2.$

Take the expression and multiply it by itself three times.

You need to multiply $2$ by itself three times and multiply $x^2$ by itself three times.

$\begin{aligned} & 2 \times 2 \times 2=8 \\\\ & x^2 \times x^2 \times x^2=x^6 \end{aligned}$

$\left(2 x^2\right) \times\left(2 x^2\right) \times\left(2 x^2\right)=8 x^6$

Write the cubed expression.

(2 x)^2=8 x^6

How to simplify cube roots

In order to simplify cube root expressions:

Look to see if the number or variable is a perfect cube.
If they are not perfect cubes, rewrite them with perfect cube factors.
Take the cube root of the perfect cubes.
Write the simplified answer.

Example 4: cube root of a number

Simplify $\sqrt[3]{-512}.$

Look to see if the number or variable is a perfect cube.

$512$ is a perfect cube number because $-8 \times -8 \times -8 = -512.$

If they are not perfect cubes, rewrite them with perfect cube factors.

The number is a perfect cube number.

Take the cube root of the perfect cubes.

\sqrt[3]{-512}=-8

Write the simplified answer.

$-8$ is the simplified answer.

Example 5: cube root of non perfect cube number

Simplify the expression $\sqrt[3]{54}.$

Look to see if the number or variable is a perfect cube.

$54$ is not a perfect cube.

If they are not perfect cubes, rewrite them with perfect cube factors.

$54$ does have a perfect cube factor. $54$ can be rewritten as $27 \times 2,$ where $27$ is a perfect cube.

$\begin{aligned} & \sqrt[3]{27 \times 2}=\sqrt[3]{27} \times \sqrt[3]{2} \\\\ & \sqrt[3]{27}=3 \end{aligned}$

Take the cube root of the perfect cubes.

\sqrt[3]{27} \times \sqrt[3]{2}

$\sqrt[3]{27}=3$

$\sqrt[3]{2}$ cannot take this cube root without a calculator because $2$ is not a perfect cube.

Write the simplified answer.

The simplified answer is:

$3 \sqrt[3]{2}$ (radical form) which also can be written as $3(2)^{\frac{1}{3}}$ (exponential form).

Example 6: cube root of an expression that is not a perfect cube

Simplify the expression $\sqrt[3]{8 x^7}.$

Look to see if the number or variable is a perfect cube.

$8$ is a perfect cube and $x^7$ can be rewritten to be $x^6 \times x^1.$ If the exponent is divisible by $3,$ it is considered to be a perfect cube.

If the exponent is not divisible by $3,$ rewrite the exponential expression so that one of the exponents is divisible by $3.$

If they are not perfect cubes, rewrite them with perfect cube factors.

\sqrt[3]{8 x^6 x^1}=\sqrt[3]{8} \times \sqrt[3]{x^6} \times \sqrt[3]{x^1}

Take the cube root of the perfect cubes.

\begin{aligned} & \sqrt[3]{8} \times \sqrt[3]{x^6} \times \sqrt[3]{x^1} \\\\ & \sqrt[3]{8}=2 \\\\ & \sqrt[3]{x^6}=x^2 \text { OR }\left(x^6\right)^{\frac{1}{3}}=x^2 \end{aligned}

$\sqrt[3]{x^1} \rightarrow$ this is not a perfect cube and can stay as $\sqrt[3]{x^1}$ or can be written as $x^{\frac{1}{3}}$

Write the simplified answer.

$\sqrt[3]{8 x^7}=2 x^2 \sqrt[3]{x}$ OR $2 x^2 \times x^{\frac{1}{3}}$ which can be simplified using laws of exponents to be $2 x^{\frac{7}{3}}$

Teaching tips for cube numbers and cube roots

When simplifying cube root expressions, students recall laws of exponents to help them to simplify the answer.

Build upon prior knowledge of perfect square numbers and square roots to introduce cube numbers and cube roots.

Although there are cube root calculators that students can access from their computers, have students simplify cube roots of given numbers or given expressions without a calculator to build number sense.

Practice worksheets have their place in the classroom, however, infuse game playing or scavenger hunts for students to practice problems.

Easy mistakes to make

Incorrect understanding of cube numbers
For example, thinking that $2^3$ is $2 \times 3=6$ instead of $2 \times 2 \times 2=8$

Thinking that you can not take the cube root of a negative number
For example, taking the square root of a negative number cannot be done in the real number system. However, taking the cube root of a negative number can be done in the real number system.
$\sqrt{-4}=\pm 2i$ (imaginary number)
$\sqrt[3]{-8}=-2$

Cube numbers and cube roots problems

16

64

25

144

$64$ is a perfect cube number because $4^3=4 \times 4 \times 4=64$

125 \mathrm{~mm}^3

25 \mathrm{~mm}^3

15 \mathrm{~mm}^3

20 \mathrm{~mm}^3

A cube has equal dimensions so if one side is $5 \mathrm{~mm}$ all sides are $5 \mathrm{~mm}.$

To find the the volume of a cube, you need to multiply the $\text{length} \times \text{width} \times \text{height}$ which in this case is $5 \times 5 \times 5=125.$

So the volume of the cube is $125 \mathrm{~mm}^3.$

9 x^6

-9 x^3

27 x^6

-27 x^8

$-3 x^2$ cubed is $-3 x^2 \times-3 x^2 \times-3 x^2, \; -3$ multiplied by itself three times is $-27$ and $-x^2$ multiplied by itself three times is $-x^8.$

You can apply laws of exponents when multiplying $x^2$ by itself three times.

-243

243

-9

9

$\sqrt[3]{-727}$ means to find the number that is multiplied to itself three times to get $-729.$

In this case $(-9)^3=-9 \times-9 \times-9=-729$

3 x^3

9 x^3

3 x

9 x

$\sqrt[3]{27 x^9}$ can be rewritten as $\left(27 x^9\right)^{\frac{1}{3}}. \; 27$ is a perfect cube number because

$3 \times 3 \times 3 = 27$ and $x^9$ is also a perfect cube expression because the exponent is divisible by $3.$

You can also apply law of exponents to simplify $\left(x^9\right)^{\frac{1}{3}}=x^3.$

\sqrt[3]{27 x^9}=3 x^3

8 x^2

8 x^3

2 x \sqrt[3]{2 x}

2 x \sqrt[3]{3 x}

In the expression, $\sqrt[3]{24 x^4}$ look for the factor of $24$ that is a perfect cube number. $24$ can be written as $8 \times 3.$

Expressions with exponents are perfect cubes if the exponent is divisible by $3.$ So, $x^4$ can be rewritten as $x^3 \times x.$

$\sqrt[3]{8 \times 3 \times x^3 \times x}$ , this is the same as $\sqrt[3]{8} \; \sqrt[3]{3} \; \sqrt[3]{x^3} \; \sqrt[3]{x}$ where $8$ and $x^3$ are perfect cube expressions.

So, $\sqrt[3]{8}=2, \sqrt[3]{x^3}=x^1$ . The other two expressions are not perfect cubes, so they remain under the cube root symbol.

\sqrt[3]{24 x^4}=2 x \sqrt[3]{3 x}

You can also use laws of exponents to simplify the answer.

Cube numbers and cube roots FAQs

What is the cube root formula?

The cube root formula can be used to represent any number in the form of its cube root. For example, for a number $x,$ the cube root is represented by $\sqrt[3]{x}=(x)^{\frac{1}{3}},$ where $x^3=x \times x \times x.$

Can you find the cube root of any number?

Yes, you can find the cube root of any number, even complex numbers.

Does the prime factorization method help when finding the cubic root of any number?

The prime factorization method can help when finding the cube root of the original number. However, knowing perfect cube numbers can be more helpful.

Can you find any root of a number?

Yes you can find any root of a number, (the nth root). You might need a calculating device to figure it out.

The next lessons are

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