Radicals - Math Steps, Examples & Questions

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Request a demo

In order to access this I need to be confident with:

Factors Fractions Square root Cube root

Math resources Number and quantity Exponents

Radicals

Here you will learn about roots which are also known as radicals. You will learn how to simplify radicals as well as how to express them as exponential expressions with fractional exponents.

Students first learn about radicals such as square roots and cube roots in $8$ th grade math and expand that knowledge as they progress through algebra $I$ and algebra $II.$

What are radicals?

Radicals (or sometimes referred to as surds or roots) are represented by $\sqrt{\;\;}$ and are used to calculate the square root or the nth root of numbers and expressions.

$\sqrt{\;\;} \rightarrow$ radical symbol

Expressions with $\sqrt{\;\;}$ are called radical expressions.

$\sqrt{16} \rightarrow$ “radical $16$ ” or “square root of $16$ ”

$\sqrt{16}$ can be simplified because $16$ is a perfect square number.

\sqrt{16}=4

All perfect square numbers have whole number roots.

Take a look at the table below, the square roots are whole numbers.

When the number under the radical or square root symbol is not a perfect square number, look to simplify it.

For example, $\sqrt{8}$ is not a perfect square number so one way to find the root is to use a calculator. $\sqrt{8} \approx 2.82843 \ldots$

Another strategy is to simplify the number under the radical by breaking the number into perfect square factors. There is a perfect square factor of $8$ which is $4.$

$\sqrt{8}=\sqrt{4} \times \sqrt{2} \rightarrow$ You can take the square root of $4$

\begin{aligned}&=2 \times \sqrt{2} \\\\ &=2 \sqrt{2} \end{aligned}

$\sqrt{8}=2 \sqrt{2} \rightarrow$ This is the simplified radical expression.

$\sqrt{5} \rightarrow$ You cannot break $5$ apart into perfect square factors because it does not have perfect square factors.

So, you can leave $\sqrt{5}$ as is or get the decimal approximation.

\sqrt{5} \approx 2.23606 \ldots

If the number under the radical is not a perfect square number, it is considered to be irrational because it is a non-terminating, non-repeating decimal. If you don’t have a calculator, you can look to simplify it or leave it as is.

Check out the webpage on square roots to get more explanations and examples.

See also: Square root

See also: Simplifying radicals

[FREE] Exponents Check for Understanding Quiz (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of exponents. 15+ questions with answers covering a range of 4th and 6th grade exponents topics to identify areas of strength and support!

DOWNLOAD FREE

[FREE] Exponents Check for Understanding Quiz (Grade 4 to 6)

DOWNLOAD FREE

If the radical is a cube root, you can work with it similarly to the way you work with a square root.

For example, $\sqrt[3]{8}$ “cube root of $8$ ” is a perfect cube number because $2 \times 2 \times 2=8$

So, the $\sqrt[3]{8}=2$

The cube roots of perfect cube numbers are integers.

Take a look at the table below:

If the number under the cube root is not a perfect cube, you can get a decimal approximation which is an irrational number, or if you do not have a calculator you can apply the same strategy to simplify it.

$\sqrt[3]{24} \rightarrow 24$ is not a perfect cube number but it has a perfect cube factor which is $8.$

You can break apart $24$ into its perfect cube factor.

$\sqrt[3]{24}=\sqrt[3]{8} \times \sqrt[3]{3} \rightarrow$ You can now take the cube root of $8.$

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

$\sqrt[3]{24}=2 \sqrt[3]{3} \rightarrow$ Simplified expression

Check out the webpage on cube roots to get more explanations and examples.

See also: Cube root

There are other radicals other than square roots and cube roots.

$\sqrt[4]{\;\;} \, \rightarrow \, 4$ th root

$\sqrt[5]{\;\;} \, \rightarrow \, 5$ th root

$\sqrt[6]{\;\;} \, \rightarrow \, 6$ th root

Etc…

Radical expressions can be rewritten as exponential expressions because all roots translate into a rational power (fraction).

For example, take a look at the table below to see some of the translations.

The index of the radical is the little number in the radical symbol.

In the example below, the index is $4.$

So, you can rewrite $\sqrt{25}$ to be $(25)^{\frac{1}{2}}$ which you can simplify to be $5$ .

\sqrt{25}=(25)^{\frac{1}{2}}=5

Similarly, $\sqrt[3]{27}$ can be rewritten to be $(27)^{\frac{1}{3}}$ which can be simplified to be $3.$

\sqrt[3]{27}=(27)^{\frac{1}{3}}=3

However, what if the expression is $\sqrt{3^3},$ can it be translated into an exponential expression? The answer is yes, it can.

Since the root is a square root, the denominator of the fraction exponent is a $2.$ The expression under the radical is $3^3,$ the power is $3,$ the power of the expression under the radical is the numerator of the fraction exponent.

So, you can rewrite $\sqrt{3^3}$ to be, $3^{\frac{3}{2}}.$

What are radicals?

Common Core State Standards

How does this apply to $8$ th grade 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, $32 \times 3–5=3–3=1/33 = 1/27.$

Grade 8 – Expressions and Equations: (8.EE.A.2)
Use square root and cube root symbols to represent solutions to equations of the form $x^{2}=p$ and $x^{3}=p,$ 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.

How to rewrite radicals to fractional exponents

In order to rewrite radicals as a fractional exponent:

Identify the index of the radical.
Identify if the expression has a power.
Make the fraction exponent have the index as the denominator and the power as the numerator.
Rewrite the expression.

Rewriting radical examples

Example 1: rational exponents

Rewrite the radical expression, $\sqrt{6}$ , as an exponential expression.

Identify the index of the radical.

The index of $\sqrt{6}$ is $2.$ All square roots have an index of $2.$

2Identify if the expression has a power.

The expression under the radical has a power of $1$ because $6$ and $6^1$ are the same.

3Make the fraction exponent have the index as the denominator and the power as the numerator.

The numerator of the fractional exponent is $1$ , and the denominator of the fraction exponent is $6.$

4Rewrite the expression.

\sqrt{6}=6^{\frac{1}{2}}

Example 2: rational exponents

Rewrite the radical expression, $\sqrt[3]{5^2}$ , as an exponential expression.

Identify the index of the radical.

The index of the radical is $3.$

Identify if the expression has a power.

The expression under the radical has a power of $2.$

Make the fraction exponent have the index as the denominator and the power as the numerator.

The fractional exponent has a numerator of $2$ and a denominator of $3.$

Rewrite the expression.

\sqrt[3]{5^2}=5^{\frac{2}{3}}

How to simplify radicals that are square roots

In order to simplify radicals that are square roots:

Find the largest square factor of the number under the radical.
Break the number under the radical into its perfect square factors.
Simplify the radical.

Simplifying radical examples

Example 3: simplifying square roots

Simplify the radical, $\sqrt{100}.$

Find the largest square factor of the number under the radical.

$100$ is a perfect square factor.

Break the number under the radical into its perfect square factors.

You do not need to break $100$ apart because it is a perfect square number.

Simplify the radical.

\sqrt{100}=10

Example 4: simplifying radical expressions

Simplify the radical, $\sqrt{32}.$

Find the largest square factor of the number under the radical.

The largest square factor of $32$ is $16.$

Break the number under the radical into its perfect square factors.

\sqrt{32}=\sqrt{16} \times \sqrt{2}

Simplify the radical.

\begin{aligned}&\begin{aligned}\sqrt{32} & =\sqrt{16} \times \sqrt{2} \\\\ & =4 \times \sqrt{2} \\\\ & =4 \sqrt{2} \end{aligned} \\\\ &\sqrt{32}=4 \sqrt{2} \end{aligned}

How to simplify radicals that are cube roots

In order to solve radical equations:

Find the largest perfect cube factor of the number under the radical.
Break the number under the radical into its perfect cube factors.
Simplify the radical.

Example 5: simplifying radical expressions

Simplify the radical, $\sqrt[3]{64}.$

Find the largest perfect cube factor of the number under the radical.

$64$ is a perfect cube number, so you do not need to find the perfect cube factors.

Break the number under the radical into its perfect cube factors.

You do not need to break the number apart into its perfect cube factors.

Simplify the radical.

\sqrt[3]{64}=4

Example 6: simplifying radical expressions

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

Find the largest perfect cube factor of the number under the radical.

The largest perfect cube factor of $54$ is $27.$

Break the number under the radical into its perfect cube factors.

\sqrt[3]{54}=\sqrt[3]{27} \times \sqrt[3]{2}

Simplify the radical.

\begin{aligned}\sqrt[3]{54} & =\sqrt[3]{27} \times \sqrt[3]{2} \\\\ & =3 \times \sqrt[3]{2} \\\\ & =3\sqrt[3]{2} \\\\ \sqrt[3]{54} & =3 \sqrt[3]{2} \end{aligned}

Teaching tips for radicals

Incorporate discovery based learning activities so that students can be like mathematicians and explore how to simplify radicals.

If students struggle with multiplication tables, have them use a digital, online square root calculator.

Reinforce perfect square numbers and perfect cube numbers with students.

Use digital platforms such as Desmos to help students develop conceptual understanding.

Have students practice problems by using interactive digital platforms such as Khan Academy.

Easy mistakes to make

Thinking that the square root is dividing a number by $\bf{2}$
For example, $\sqrt{16}$ ≠ $8$

$\sqrt{16}=4$ because $4 \times 4=16$

Rewriting the number under the square root sign (the radicand) as a product of any two factors
In order to simplify a radical, it has to be a perfect square or has to have a perfect square factor. So, one of these factors must be a square number.

To simplify $\sqrt{40},$ break $40$ so that one of the factors is a perfect square. The largest square factor of $40$ is $4.$

$\begin{aligned}\sqrt{40} & =\sqrt{4} \times \sqrt{10} \\\\ & =2 \times \sqrt{10} \\\\ & =2 \sqrt{10} \end{aligned}$

Thinking that the index of the radical is the power of the exponent
For example, thinking that $\sqrt[3]{4}=(4)^3,$ which is not a true statement,

$\sqrt[3]{4}$ ≠ $(4)^3$

Radical expressions can be written as exponential expressions with fractional exponents.

Practice radical questions

121^1

(121)^{\frac{1}{2}}

(121)^2

(121)^{\frac{1}{3}}

Radical expressions can be rewritten as exponential expressions with fractional exponents. $“\sqrt{ \;\;}”$ is a square root radical with an index of $2.$

The index is the denominator of the fractional exponent and the power of the expression is the numerator, $121=121^1$ so the numerator is $1$ and the denominator is $2.$

\sqrt{121}=(121)^{\frac{1}{2}}

(5)^{\frac{1}{3}}

(5)^{\frac{1}{4}}

(5)^{\frac{4}{3}}

(5)^{\frac{3}{4}}

Rewriting the radical expression as an exponential expression with a fractional exponent.

The index of the radical is the denominator of the fractional exponent and the power of the expression is the numerator of the fraction.

In this case, the index of the radical is $4$ and the power of the expression is $3.$

\sqrt[4]{5^3}=(5)^{\frac{3}{4}}

x^{\frac{5}{2}}

x^5

x^{\frac{2}{5}}

x^2

Rewriting the radical expression as an exponential expression with a fractional exponent (rational exponent).

The index of the radical is the denominator of the fractional exponent and the power of the expression is the numerator of the fraction.

In this case, the index is $2$ and the power is $5.$

\sqrt{x^5}=x^{\frac{5}{2}}

3 \sqrt{4}

4 \sqrt{3}

2 \sqrt{3}

3 \sqrt{2}

To simplify a radical expression that is a square root, look for the largest perfect square factor of the number. In this case, the largest perfect square factor of $12$ is $4.$

Rewrite the number in the radical with its perfect square factor.

\begin{aligned}\sqrt{12} & =\sqrt{4} \times \sqrt{3} \\\\ & =2 \times \sqrt{3} \\\\ & =2 \sqrt{3} \\\\ \sqrt{12} & =2 \sqrt{3} \end{aligned}

4 \sqrt{5}

16 \sqrt{5}

5 \sqrt{4}

2 \sqrt{5}

To simplify a radical expression that is a square root, look for the largest perfect square factor of the number. In this case, the largest perfect square factor of $80$ is $16.$

Rewrite the number in the radical with its perfect square factor.

\begin{aligned}\sqrt{80} & =\sqrt{16} \times \sqrt{5} \\\\ & =4 \times \sqrt{5} \\\\ & =4 \sqrt{5} \end{aligned}

\sqrt{80}=4 \sqrt{5}

5 \sqrt{2}

2 \sqrt[3]{5}

5 \sqrt[3]{2}

2 \sqrt{5}

To simplify a radical expression that is a cube root, look for the largest perfect cube factor of the number. In this case, the largest perfect cube factor of $40$ is $8.$

Rewrite the number in the radical with its perfect cube factor.

\begin{aligned}\sqrt[3]{40} & =\sqrt[3]{8} \times \sqrt[3]{5} \\\\ & =2 \times \sqrt[3]{5} \\\\ & =2 \sqrt[3]{5} \end{aligned}

\sqrt[3]{40}=2 \sqrt[3]{5}

Radicals FAQs

How can you multiply radicals?

Multiplying radicals is a concept you will learn in algebra $I$ and algebra $II.$ However, when multiplying radicals, you multiply the numbers outside the radical together and the numbers under the radical together. Be sure to simplify the radical if needed.

How can you add and subtract radicals?

Adding and subtracting radicals is a concept you will learn in algebra $I$ and algebra $II.$ However, when adding or subtracting radicals, you have to make sure the number under the radicals are the same or you cannot add/subtract the expressions.

How can you divide radical expressions?

You learn about dividing radical expressions in algebra $I$ and algebra $II.$ However, dividing a radical expression (finding the quotient), write the radical division as a rational expression (numerator and denominator) and then divide the numbers under the radical.

If the numbers do not divide, rationalize the denominator. In more complex problems, you will have to multiply the expression by the conjugate.

Can you graph radical expressions?

Yes, radical functions can be graphed like any other function. In algebra $I$ and algebra $II,$ you will learn how to graph all types of functions such as radical equations, logarithm equations, quadratic equations, polynomial equations, linear equations, and absolute value equations.

Can you solve radical equations?

Yes, you can solve radical equations.

How do you rewrite a radical expression as an exponential function if there is a number in front of the radical?

If there is a number in front of the radical (coefficient of the radical), leave it there and just rewrite the number under the radical with the fractional exponent. For example, $3 \sqrt{7}=3(7)^{\frac{1}{2}}.$

What are imaginary numbers?

Imaginary numbers are part of the complex number system. If you try to take the square root of a negative number on the calculator, you will get an “error.” In algebra $II,$ you will learn that the root of a negative number is an imaginary number.

How can you solve radical equations?

Solving radical equations is similar to the way you solve other equations. You will use the inverse operation which is to square both sides

The next lessons are

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.