Cube Root of Perfect Cubes — Examples & Practice

Simplifying Cube Roots - Perfect Cubes and Harder Examples

by Math Meeting

This video by Math Meeting goes over simplification of cube roots.

$\large{\text{Summary}}$

Cube roots are simplified similarly to square roots. We need to find what number can be multiplied by itself $3$ times to make the perfect cube under the radical.

The first example is with $\sqrt[3]{8}$. First, we're going to factor out $8$. $$\begin{align}\sqrt[3]{8} &= \sqrt[3]{4\cdot 2}\\ &= \sqrt[3]{2\cdot2\cdot2}\end{align}$$ We can see that $8$ evenly breaks into $3$ factors of $2$. This means that the cube root of $8$ is $2$. $$\sqrt[3]{8} = 2$$

A big difference between square roots and cube roots are that perfect cubes can be negative. This is because the product of an odd number of negatives is negative. Let's look at $\sqrt[3]{(-125)}$. $$\begin{align}\sqrt[3]{(-125)} &= \sqrt[3]{25(-5)}\\ &=\sqrt[3]{(-5)(-5)(-5)}\end{align}$$ We have split $-125$ into $3$ of the same factors, so we have the simplified version of our cube root. $$\sqrt[3]{(-125)}= -5$$

Finding the Cube Root of a Perfect Cube

Say you have a perfect cube, $x^3.$ If you take the cube root of $x^3,$ you get: $$\sqrt[3]{x^3} = x$$

Here are some examples:

$$\sqrt[3]{8} = \sqrt[3]{2 \cdot 2 \cdot 2} = \sqrt[3]{2^3} = 2$$ $$\sqrt[3]{27} = \sqrt[3]{3 \cdot 3 \cdot 3} = \sqrt[3]{3^3} = 3$$ $$\sqrt[3]{\dfrac{1}{64}} = \sqrt[3]{\dfrac{1}{4} \cdot \dfrac{1}{4} \cdot \dfrac{1}{4}} = \sqrt[3]{\bigg(\dfrac{1}{4}\bigg)^3} = \dfrac{1}{4}$$

It turns out all the examples above involve positive numbers. There are also examples with negative numbers. Here's one of them.

$$\sqrt[3]{-8} = \sqrt[3]{-2 \cdot -2 \cdot -2} = \sqrt[3]{(-2)^3} = -2$$

The tricky thing to keep track of is all the negative signs. Here is a quick check for yourself:

The cube root of a positive number is positive.

The cube root of a negative number is negative.

Cube Root of a Perfect Cube

Recall that a perfect cube is the product of an integer multiplied by itself three times. $$\text{Perfect Cube}=a\times a\times a$$

Using exponents, we can rewrite this as: $$\text{Perfect Cube}=a^3$$

For example, $3\times3\times3=27$. So, we know $27$ is a perfect cube. Writing this using exponents, gives us: $$27=3^3$$

Radicals, or roots, are the inverse operation of exponents. In other words, they undo an exponent. If you square a number, and then take the square root, you return back to that number. Similarly, if you cube a number, then take that cube root, you return back to the number. $$3^3=27\quad\rightarrow\quad\sqrt[3]{27}=3$$

So, the cube root of a perfect cube will be the integer that's being multiplied by itself three times. $$\begin{array}{ccc} 1\times1\times1=1&&\sqrt[3]{1}=1\\ 2\times2\times2=8&&\sqrt[3]{8}=2\\ 3\times3\times3=27&&\sqrt[3]{27}=3\\ 4\times4\times4=64&&\sqrt[3]{64}=4\\ 5\times5\times5=125&&\sqrt[3]{125}=5\\ 6\times6\times6=216&&\sqrt[3]{216}=6\\ 7\times7\times7=343&&\sqrt[3]{343}=7\\ 8\times8\times8=512&&\sqrt[3]{512}=8\\ 9\times9\times9=729&&\sqrt[3]{729}=9\\ 10\times10\times10=1{,}000&&\sqrt[3]{1{,}000}=10 \end{array}$$

There are more perfect cubes, of course. But they become difficult to calculate by hand when the base gets large.

Finding the Cube Root of a Perfect Cube

Recall that a perfect cube is the number that is the result of multiplying a number with itself $3$ times.

We can think of cube roots in the same context that we view square roots. When we take the square root of a perfect square, we are searching for the number, that when multiplied by itself two times, results in the perfect square. Similarly, when we are finding the cube root of the perfect cube, we are searching for the number, that when multiplied by itself three times, results in the perfect cube. Let's walk through an example.

Find $\sqrt[3]{343}$

To find this, we first need to break $343$ into its prime factorization. To do so, we need to find the first pair of factors that include a prime number. For $343$, this first pair will be $7$ and $49$. $7$ cannot be broken down any further, but $49$ can be broken into $7$ and $7$. Therefore, we can say that $\sqrt[3]{343} = \sqrt[3]{7 \cdot 7 \cdot 7}$, so we can say that the cube root of $343$ is $7$, where $7 \cdot 7 \cdot 7 = 343$.