Cube Roots

The cube root of a number $a$ is the answer to the question, "What number, when cubed $($raised to the 3$^\text{rd}$ power$)$, results in $a?$" The symbol for cube root is "$\sqrt[3]{\ }$". The cube root of the number $a$ is written as $\sqrt[3]{a}$.

What is the cube root of $64$?

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Ask yourself the question, "What number, when cubed, results in $64$?" The answer to that question will be the cube root of $64$.

$4^3=64$, so $\sqrt[3]{64}=4$. $_\square$

The cube root is often used to solve cubic equations. In particular, it can be used to solve for the dimensions of a three-dimensional object of a certain volume.

The cube root of a number $a$, denoted as $\sqrt[3]{a},$ is the number $b$ such that

$$b^3=a.$$

The cube root symbol acts similarly to the square root symbol. It is often called a radical, and the number or expression underneath the top line of the symbol is called the radicand. The cube root symbol is a grouping symbol, meaning that all operations in the radicand are grouped as if they were in parentheses.

Unlike a square root, the result of a cube root can be any real number: positive, negative, or zero. Also different from a square root is the domain restriction on the radicand: the radicand of a cube root can be negative while still achieving a real result for the cube root.

To solve for the cube root of any integer, first ask yourself the question, "What integer, when cubed, results in this number?" If none comes to mind, list perfect cubes until a match for the radicand is found.

What is the value of $\sqrt[3]{216}$?

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ANSWER

Think of perfect cubes until you find a match for the radicand.

$6^3=216$, so $\sqrt[3]{216}=6$. $_\square$

The process is similar for the cube roots of fractions. Look for perfect cubes that match the numerator and denominator of the fraction.

What is the value of $\sqrt[3]{\dfrac{27}{125}}.$

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ANSWER

$3^3=27$, and $5^3=125$.

It follows that $\sqrt[3]{\dfrac{27}{125}}=\dfrac{3}{5}.$ $_\square$

Unlike the square root, the cube root has no domain restriction under the real numbers. The radicand can be any real number, and the result of the cube root will be a real number.

What is the value of $\sqrt[3]{-8}$?

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ANSWER

Similarly to previous examples, the cube root of $-8$ is the answer to the question, "What number, when cubed, results in $-8$?"

$(-2)^3=-8$, so it follows that $\sqrt[3]{-8}=-2.\ _\square$

In general, if a cube root operation is done on a negative number, then the result is negative.

Let $a$ be a real number. Then,

$$\sqrt[3]{-a}=-\sqrt[3]{a}.$$

The process to simplify cube roots of non-perfect cubes is like the process to simplify square roots.

Let $a$ be a non-perfect cube integer.

The simplified radical form of the cube root of $a$ is

$$b\sqrt[3]{c}.$$

In this form, $\sqrt[3]{a}=b\sqrt[3]{c}$, $b$ and $c$ are integers, and $c$ is positive with no perfect cube factors other than $1$.

To simplify a cube root, first look for the largest perfect cube factor of the radicand. Then, apply the following property:

Let $a$ and $b$ be real numbers. Then,

$$\sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}.$$

Simplify $\sqrt[3]{81}.$

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ANSWER

The goal is to find the largest perfect cube factor of $81$. Since $27$ is that factor, we have

$$\begin{aligned} \sqrt[3]{81}&=\sqrt[3]{27\times 3} \\ &=\sqrt[3]{27}\times\sqrt[3]{3} \\ &=3\sqrt[3]{3}.\ _\square \end{aligned}$$

Note: When a number is placed to the left of a cube root symbol, multiplication is implied. Therefore, "$3\sqrt[3]{3}$" is read as "$3$ times the cube root of $3$."

The cube root of a complex number is somewhat ambiguous. Non-real complex numbers are neither positive nor negative, so it is not well-defined which cube root is the principal root. Therefore, when a cube root operation is done on a complex number, the result is interpreted to be all solutions of an equation:

Let $z$ be a complex number. Then there are up to three values for $\sqrt[3]{z}$, and they are equal to the solutions of the equation

$$x^3=z.$$

Note that the cube root operation, when used on complex numbers, is not well-defined in the sense that there is likely more than one result.

The process for finding the cube roots of a complex number is similar to the process for finding the $3^\text{rd}$ roots of unity.

What are the values of $\sqrt[3]{i}$?

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ANSWER

First, it is necessary to write the complex number in polar form. For $i$, $r=|i|=1$ and $\theta=\text{arccot}\left(\frac{0}{1}\right)=\frac{\pi}{2}+2k\pi$, where $k$ is an integer.

$$\begin{array}{c} i & = & e^{i\pi/2} & = & e^{i5\pi/2} & = & e^{i9\pi/2} \\ \sqrt[3]{i} & = & i^{1/3} \\\\ \sqrt[3]{i} & = & \left(e^{i\pi/2}\right)^{1/3} & = & e^{i\pi/6} & = & \frac{\sqrt{3}}{2}+\frac{i}{2} \\ \sqrt[3]{i} & = & \left(e^{i5\pi/2}\right)^{1/3} & = & e^{i5\pi/6} & = & -\frac{\sqrt{3}}{2}+\frac{i}{2} \\ \sqrt[3]{i} & = & \left(e^{i9\pi/2}\right)^{1/3} & = & e^{i3\pi/2} & = & -i. \end{array}$$

There are three possible results for $\sqrt[3]{i}$: $\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}$, $-\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}$, and $-i$. $_\square$