What is a radical in math?

A radical is the mathematical inverse operation of an exponent. For example, a square root (radical with index 2) undoes the operation of squaring a number.

The radicand is the number or expression located inside the radical symbol.

Do radicals and powers cancel each other out?

Yes, a radical and a power with the same index undo each other. Applying both operations successively returns the original input value.

What Are Radicals in Math? Definitions & Practice Questions

Understanding the Power of $n$

To understand radicals, we first need to review powers (or exponents). Let the power of $2$ operation be represented by the diagram below:

More examples of the power $2$ operation:

Input = $4 \implies$ Output = $4^2 = 4 \times 4 = 16$
Input = $10 \implies$ Output = $10^2 = 10 \times 10 = 100$

Similarly, the diagram below represents the power $3$ operation:

More examples of the power $3$ operation:

Input = $3 \implies$ Output = $3^3 = 3 \times 3 \times 3 = 27$
Input = $1 \implies$ Output = $1^3 = 1 \times 1 \times 1 = 1$
Input = $4 \implies$ Output = $4^3 = 4 \times 4 \times 4 = 64$

General Power of $n$:

$$a^n = a \times a \times a \dots \times a \quad \text{(multiplied } n \text{ times)}$$

Definition of Radicals

1. Radical with index 2 (Square Root)

The square root is the inverse operation of the power $2$. We interchange the input and output. For example, if the input is $9$, the output of the inverse operation is $3$.

Diagram showing the radical with index 2 (square root) operation

We write this as:

$$\sqrt[2]{9} = 3 \quad \text{because} \quad 9 = 3^2$$

Input = $16 \implies$ Output = $\sqrt{16} = 4 \quad \text{(because } 4^2 = 16\text{)}$
Input = $25 \implies$ Output = $\sqrt{25} = 5 \quad \text{(because } 5^2 = 25\text{)}$

Terminology: The symbol $\sqrt{\phantom{x}}$ is the radical. The number $2$ is the index. The number inside the radical is the radicand.

Diagram pointing out the radical symbol, the radicand, and the index

Note: By convention, a radical with an index of $2$ (square root) is written without the $2$. We simply write $\sqrt{\phantom{x}}$.

2. Radical with index 3 (Cube Root)

The inverse operation of the power $3$ operation is called the cube root.

Diagram showing the radical with index 3 (cube root) operation

We write this as:

$$\sqrt[3]{8} = 2 \quad \text{because} \quad 8 = 2^3$$

Input = $27 \implies$ Output = $\sqrt[3]{27} = 3 \quad \text{(because } 3^3 = 27\text{)}$
Input = $125 \implies$ Output = $\sqrt[3]{125} = 5 \quad \text{(because } 5^3 = 125\text{)}$

3. General Radical with index $n$ ($n^{th}$ Root)

We can generalize radicals with any index $n$ (where $n$ is a whole number greater than 0):

$$\text{If } y = a^n, \text{ then } \sqrt[n]{y} = a$$

More Examples:

$y = 3^5 = 243$, therefore $\sqrt[5]{243} = 3$
$y = 10^6 = 1,000,000$, therefore $\sqrt[6]{1,000,000} = 10$
$y = (-2)^3 = -8$, therefore $\sqrt[3]{-8} = -2$
$y = 1^{20} = 1$, therefore $\sqrt[20]{1} = 1$ (In general, $\sqrt[n]{1} = 1$)
$y = 0^9 = 0$, therefore $\sqrt[9]{0} = 0$ (In general, $\sqrt[n]{0} = 0$)

IMPORTANT RESTRICTION: These relationships are NOT valid in real numbers if $n$ is an EVEN integer and $y$ is NEGATIVE.

$\sqrt{-16}$ is undefined in real numbers because no real number $x$ exists such that $x^2 = -16$. (The square of a real number is always positive or zero).
$\sqrt[4]{-1}$ is undefined in real numbers for the same reason.

Radicals and Powers Undo Each Other

Radicals and power operations with the same index undo each other. Because they are inverse operations, applying both successively returns the original input.

Example: The square root undoes power $2$

Diagram showing that taking the square root of a squared number returns the original number

Written mathematically: $$\sqrt{3^2} = 3$$

Example: Power $2$ undoes the square root

Diagram showing that squaring a square root returns the original number

Written mathematically: $$(\sqrt{9})^2 = 9$$

More Examples:

$(\sqrt{12})^2 = 12$
$\sqrt{8^2} = 8$
$(\sqrt[3]{5})^3 = 5$
$(\sqrt[3]{10})^3 = 10$

In general, for $a \ge 0$, we can write:

$$\sqrt[n]{a^n} = a \quad \text{and} \quad (\sqrt[n]{a})^n = a$$

Practice Questions

Note: Do NOT use a calculator to answer the following questions.

Given the following powers:
$2^6 = 64$, $\; 3^5 = 243$, $\; 5^3 = 125$, $\; 0^7 = 0$, $\; 1^{20} = 1$, $\; 2^9 = 512$, $\; 5^5 = 3125$, $\; 10^5 = 100000$, $\; 0.1^3 = 0.001$

Find the exact values of the following radicals:
$\sqrt[9]{512}$, $\; \sqrt[5]{3125}$, $\; \sqrt[5]{243}$, $\; \sqrt[6]{64}$, $\; \sqrt[3]{0.001}$, $\; \sqrt[20]{1}$, $\; \sqrt[5]{100000}$, $\; \sqrt[7]{0}$, $\; \sqrt[3]{125}$
View Solution
- $\sqrt[9]{512} = \mathbf{2} \quad$ (because we are given $2^9 = 512$)
- $\sqrt[5]{3125} = \mathbf{5} \quad$ (because $5^5 = 3125$)
- $\sqrt[5]{243} = \mathbf{3} \quad$ (because $3^5 = 243$)
- $\sqrt[6]{64} = \mathbf{2} \quad$ (because $2^6 = 64$)
- $\sqrt[3]{0.001} = \mathbf{0.1} \quad$ (because $0.1^3 = 0.001$)
- $\sqrt[20]{1} = \mathbf{1} \quad$ (because $1^{20} = 1$. Note: $\sqrt[n]{1} = 1$ for any $n > 0$)
- $\sqrt[5]{100000} = \mathbf{10} \quad$ (because $10^5 = 100000$)
- $\sqrt[7]{0} = \mathbf{0} \quad$ (because $0^7 = 0$. Note: $\sqrt[n]{0} = 0$ for any $n > 0$)
- $\sqrt[3]{125} = \mathbf{5} \quad$ (because $5^3 = 125$)
Given the following radicals:
$\sqrt{64} = 8$, $\; \sqrt[5]{7776} = 6$, $\; \sqrt[3]{1000} = 10$, $\; \sqrt[7]{128} = 2$, $\; \sqrt[7]{0.0000001} = 0.1$, $\; \sqrt{10000} = 100$, $\; \sqrt[4]{20736} = 12$, $\; \sqrt[9]{512} = 2$

Find the values of the following powers:
$2^7$, $\; 0.1^7$, $\; 6^5$, $\; 8^2$, $\; 2^9$, $\; 12^4$, $\; 10^3$, $\; 100^2$
View Solution
- $2^7 = \mathbf{128} \quad$ (because we are given $\sqrt[7]{128} = 2$)
- $0.1^7 = \mathbf{0.0000001} \quad$ (because $\sqrt[7]{0.0000001} = 0.1$)
- $6^5 = \mathbf{7776} \quad$ (because $\sqrt[5]{7776} = 6$)
- $8^2 = \mathbf{64} \quad$ (because $\sqrt{64} = 8$)
- $2^9 = \mathbf{512} \quad$ (because $\sqrt[9]{512} = 2$)
- $12^4 = \mathbf{20736} \quad$ (because $\sqrt[4]{20736} = 12$)
- $10^3 = \mathbf{1000} \quad$ (because $\sqrt[3]{1000} = 10$)
- $100^2 = \mathbf{10000} \quad$ (because $\sqrt{10000} = 100$)
Simplify the following expressions:
$\sqrt{5^2}$, $\; (\sqrt[5]{3})^5$, $\; \sqrt[3]{10^3}$, $\; (\sqrt[7]{128})^7$
View Solution
- $\sqrt{5^2} = \mathbf{5} \quad$ (Because the square root and the power of $2$ undo each other)
- $(\sqrt[5]{3})^5 = \mathbf{3} \quad$ (Because the radical of index $5$ and the power of $5$ undo each other)
- $\sqrt[3]{10^3} = \mathbf{10} \quad$ (Because the cube root and the power of $3$ undo each other)
- $(\sqrt[7]{128})^7 = \mathbf{128} \quad$ (Because the radical of index $7$ and the power of $7$ undo each other)

What Are Radicals in Math?