Simplifying Cube Roots - Examples & Practice

Simplifying Cube Roots

There are many different techniques you can use to simplify cube roots, similar to how you can simplify square roots. In this explanation, we will learn a slightly different method. Don't worry, you'll get the same answer either way.

Step One:

Rewrite the number or expression under the cube root in terms of its factors. You may even want to break these factors down into THEIR factors...this might help you later. For example, you might do this: $$\begin{align}\sqrt[3]{72}&=\sqrt[3]{12 \cdot 6}\\ &=\sqrt[3]{(4 \cdot 3) \cdot (2\cdot 3)}\\ &=\sqrt[3]{((2 \cdot 2) \cdot 3)\cdot (2\cdot 3)}\\ &=\sqrt[3]{2\cdot 2\cdot 2\cdot 3\cdot 3} \end{align}$$

Step Two:

Since we are taking the cube root $($i.e., $\sqrt[3]{\phantom{98}}),$ we want to look for perfect cube factors. Every time we see a group of three of the same factors, we write one copy of that factor outside of the cube root sign. Here's how it works:

$$\begin{align} \sqrt[3]{2\cdot 2\cdot 2\cdot 3\cdot 3} & \rightarrow \sqrt[3]{\boxed{2\cdot 2\cdot 2}\cdot 3\cdot 3},\text{ where 2 is the perfect cube factor}\\ &\rightarrow \boxed{2} \sqrt[3]{\boxed{2\cdot 2\cdot 2}\cdot 3\cdot 3} \end{align}$$

Step Three:

Next, remember the groups of three perfect cube factors we found under the cube root? Eliminate them from under the cube root, like this: $$\begin{align} \boxed{2} \sqrt[3]{\boxed{2\cdot 2\cdot 2}\cdot 3\cdot 3} &\rightarrow \boxed{2} \sqrt[3]{\cancel{2\cdot 2\cdot 2}\cdot 3\cdot 3}\\ &\rightarrow 2 \sqrt[3]{3 \cdot 3} \end{align}$$

This result is equivalent to the cube root we are trying to simplify! In this example, $\sqrt[3]{72} = 2 \sqrt[3]{3\cdot 3}.$

Step Four:

Simplify everything! $$\begin{align} \sqrt[3]{72} &=2 \sqrt[3]{3 \cdot 3}\\ &=2 \sqrt[3]{9} \end{align}$$

Here's a graphic that goes through the steps, but with a different expression underneath the cube root!

(It is important to note that this method applies to monomials. You cannot split up radicals with sums and differences in the same way.)

Let's do another example!

Step One: Break down the expression into its factors

$$\sqrt[3]{27x^4y^2z^3} = \sqrt[3]{3 \cdot 3 \cdot 3 \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot z \cdot z \cdot z}$$

Step Two: Every time you see a pair of the same factors, write one copy of that factor outside the radical

$$\begin{align} & \sqrt[3]{3 \cdot 3 \cdot 3 \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot z \cdot z \cdot z} \\ &\rightarrow \sqrt[3]{{\boxed{3 \cdot 3 \cdot 3}} \cdot {\boxed{x \cdot x \cdot x \cdot x}} \cdot {\boxed{y \cdot y }}\cdot {\boxed{z \cdot z \cdot z}}} \\ &\rightarrow 3 \cdot x \cdot z \sqrt[3]{3 \cdot 3 \cdot 3 \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot z \cdot z \cdot z} \end{align}$$

Step Three: Eliminate the pairs from under the radical

$$\begin{align} &3 \cdot x \cdot z \sqrt[3]{\boxed{3 \cdot 3 \cdot 3} \cdot \boxed{x \cdot x \cdot x} \cdot x \cdot y \cdot y \cdot \boxed{z \cdot z \cdot z}} \\ &\rightarrow 3 \cdot x \cdot z \sqrt[3]{\cancel{3 \cdot 3 \cdot 3} \cdot \cancel{x \cdot x \cdot x} \cdot x \cdot y \cdot y \cdot \cancel{z \cdot z \cdot z}}\\ &3 \cdot x \cdot z \sqrt[3]{x \cdot y \cdot y } \end{align}$$

Step Four: Simplify!

$$\begin{align} \sqrt[3]{27x^4y^2z^3}&=3 \cdot x \cdot z \sqrt[3]{x \cdot y \cdot y }\\ &=3xz\sqrt[3]{xy^2} \end{align}$$

Get the Radicand Out from Under the Radical

When we simplify roots—cube roots or square roots—we want to get as much as we can out from underneath the radical. The term for the number or variable within the radical is called the radicand. We might not be able to get all of the radicand out, but we want to try. To do this with cube roots, we need to make sure that we're breaking down the radicand so that we can find any factor that's a perfect cube (remember that when multiplying radicals, $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ ).

Let's try it. How can we simplify $\sqrt[3]{56y^4}$? Let's think of it in terms of steps:

1. Separate out the different kind of numbers (variables vs. constants).
2. Find the factors of the terms.
3. Pull the cube roots out from under the radical.
4. Consolidate everything.

Step $1$: Separate Out Different Types of Numbers.

There are two different kinds of numbers in $\sqrt[3]{56y^4}$: constants ($56$) and variables ($y^4$). Since we can't know what $y$ is just by looking at the equation, we have to deal with it separately. For now, let's visualize the equation as $\sqrt[3]{56 \times y^4}$ so that the terms are clearly separate.

Step $2$: Find the Cube Factors.

Now, we can look at $56$ and $y^4$ separately. Let's start with $y^4$.

$y^4$ does have a perfect cube factor. $y^4 = y \cdot y \cdot y \cdot y$, which we can also write as $y^3 \cdot y$. $y^3$ is a perfect square, so let's make that clear in our problem: $$\sqrt[3]{56 \cdot y^3 \cdot y}$$

Right, the factors of $56$ are $1, 2, 4, 7, 8, 14, 28,$ and $56$.

$$8 = 2 \times 2 \times 2 = 2^3$$

$8$ is a perfect cube! Let's rewrite $56$ in terms of its factors in the original problem: $$\sqrt[3]{7 \times 8 \times y^3 \times y}$$

Step $3$: Pull the Cube Roots Out From Under the Radical.

Now that we have our expression expanded so that it's easy to find the perfect cubes, we can pull their cube roots out of the radical. To do that, you have to find the cube root of each perfect cube in the equation.

Right, the cube root of $8$ is $2$. Now that we know what it is, we can rewrite our original equation as: $$2\sqrt[3]{7 \times y^3 \times y}$$

The cube root of $y^3$ is $y$. So we can write the equation as: $$2 \times y \times \sqrt[3]{7 \times y}$$ Since there aren't any more perfect cubes inside the radical, we're just about done!

Step $4$: Put Everything Back Together

Step $4$ is tidying up the expression. We can get rid of the multiplication signs that we used to make everything clearer, compressing the equation down a little bit so it's easier to read: $$\sqrt[3]{56y^4} = 2y\sqrt[3]{7y}$$ And we're done!

(Video) Simplifying Cube Roots

by Math Meeting

This by Math Meeting shows you how to simplify cubed root expressions.

$\large{\text{Summary}}$

Simplifying cube roots is similar to simplifying square roots. We want to find any factors that are perfect cubes that are a factor of our number and pull them out.

For example, suppose we want to find $\sqrt[3]{72}$. We can factor $72$ as follows: $$\sqrt[3]{72}=\sqrt[3]{8\times9}$$ $8$ is a perfect cube. Therefore, we can split it from the $9$. $$\begin{align} \sqrt[3]{8\times9}&=\sqrt[3]{8}\times\sqrt[3]{9}\\ &=2\times\sqrt[3]{9}\\ &=2\sqrt[3]{9} \end{align}$$