Radicals are expressions in the form $\sqrt[n]{x}$, which are often used instead of their corresponding decimal, as they make mathematical expressions clearer and simpler. We need to learn how to simplify them, by using certain rules and tricks.

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A radical is a mathematical expression containing a root and is denoted by $\sqrt[n]{x}=y$. Taking the root of a number is the opposite operation of applying an exponent, in the sense that

$\mathrm{If}\sqrt[n]{x} =y,\mathrm{then}{y}^{n}=x$

In this expression, n is called the index of the root, and x is called the radicand.

The index n of a root can be any positive integer (1, 2, 3, 4...), and gives the name to the radical:

• If $n=1$we have the trivial root $\sqrt[1]{x}=x$.

• If $n=2$, $\sqrt[2]{x}$ is called the square root of x. Often the index 2 is omitted, and one writes simply $\sqrt{x}$. The square root of a number x is a number y that when raised to the power of 2 gives x:

$\mathrm{If}\sqrt{x}=y,\mathrm{then}{y}^{2}=x$

• If $n=3$, $\sqrt[3]{x}$ is called the cubic root of x. The cubic root of a number x is the number y that when raised to the power of 3 gives x:

$\mathrm{If}\sqrt[3]{x}=y,\mathrm{then}{y}^{3}=x$

Which number y raised to the power of 3 gives 8? In other words, solve the equation

${y}^{3}=8$

Solution:

To do so, you need to apply the cubic root to this expression to find the value of y:

${y}^{3}=8\phantom{\rule{0ex}{0ex}}\overline{)\sqrt[3]{{y}^{\overline{)3}}}}=\sqrt[3]{8}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{y}\mathbf{=}\mathbf{2}}$

since ${2}^{3}=8$

As you can see, the exponents and radicals cancel each other out. This holds for any exponent, and it's the most useful practical rule you should keep in mind. Let's look at another example with a square root:

To solve ${x}^{2}=25$, apply the square root to find the value of x:

Solution:

${x}^{2}=25\phantom{\rule{0ex}{0ex}}\overline{)\sqrt{{x}^{\overline{)2}}}}=\sqrt{25}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{x}\mathbf{=}\mathbf{5}}$

since ${5}^{2}=25$

Exponents and radicals cancel each other out. This is the most useful practical rule you should keep in mind!

## What does "simplifying radicals" mean?

Simplifying radicals means rewriting them in the most simple and fundamental possible way.

Sometimes you'll be able to get rid of the radical symbol altogether: for example, have a look at

$\sqrt{9}=3$

On the left-hand side, we have a radical expression, while on the right-hand side an integer number. These two expressions are equal, but the one on the right-hand side is simpler, as it does not contain any radicals.

In other circumstances, you'll still have a radical, but in a simpler form than the one you started with. For example, consider

$\sqrt{8}=2\sqrt{2}$

It may not be obvious at a first glance, but also in this case the expressions on the left-hand side and the right-hand side are equal. The difference is that the expression on the right-hand side is simplified.

After reading this article, it will be easy for you to perform this kind of simplification; now, to see that these two expressions actually are the same, raise them both to the power of two:

$\begin{array}{rcc}{\left(\sqrt{8}\right)}^{2}& =& 8\\ {\left(2\sqrt{2}\right)}^{2}& =& {2}^{2}{\left(\overline{)\sqrt{2}}\right)}^{\overline{)2}}=4·2=8\end{array}$

In both cases, we obtain 8, showing that these two expressions are the same.

In order to perform the manipulations required to simplify radicals, you need to be familiar with the properties of radicals.

The properties of radicals are algebraic rules that help you manipulate radicals with the basic operations of sum, difference, product and quotient. Have a look at the article on Powers Roots And Radicals to refresh them! In particular, you should keep in mind the product rule and the quotient rule of radicals:

As long as the index of the roots is the same, you can multiply radicals with different numbers inside the root by simply combining them into one root and multiplying the numbers inside the root. Likewise, you can split a root into separate roots using factors.

$\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{a.b}$

Similarly, as long as the index of the roots is the same, you can divide radicals with different numbers inside the root by combining them into one root and dividing the numbers inside the root.

$\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

### The 3 rules to simplify radicals

The following three rules must hold for a radical to be fully simplified:

1. There are no perfect squares (except for 1) in the radicand.

2. There are no fractions in the radicand.

3. There are no radicals in the denominator of a fraction.

#### Common mistakes in simplifying radicals

It is important to be able to determine whether a radical is simplified or not, according to the rules above:

Is the radical $\sqrt{9}$ simplified?

The answer is No: the radicand contains a perfect square, 9, so by rule 1, the radical is not simplified.

Is the radical $\sqrt{\frac{1}{3}}$ simplified?

The answer is again No: the radicand contains a fraction, so by rule 2, the radical is not simplified.

Is the radical $\frac{1}{\sqrt{3}}$ simplified?

The answer is yet again No: there is a radical in the denominator of the fraction, so by rule 3, the radical is not simplified.

Let's finally see how to manipulate and simplify radicals using the properties of radicals, so that the three rules to simplify radicals hold true!

Simplify the following radical: $\sqrt{\frac{20}{9}}$.

Solution:

The radicand contains a fraction, so by rule 2. We have to get rid of it. To do so, use the quotient property to rewrite this radical as $\frac{\sqrt{20}}{\sqrt{9}}$.

Now we have to simplify separately the numerator and the denominator.

To simplify the numerator, we have to use the product property to isolate perfect squares, so that rule 1 is fulfilled. Note that $20=4·5$, so the numerator can be rewritten as

$\sqrt{20}=\sqrt{4·5}=\sqrt{4}·\sqrt{5}=2\sqrt{5}$

Now to the denominator: the square root of 9 is 3, so we can get rid of the radical in the denominator of the fraction, fulfilling rule 3, and the final result is

$\overline{)\sqrt{\frac{\mathbf{20}}{\mathbf{9}}}\mathbf{=}\frac{\mathbf{2}\sqrt{\mathbf{5}}}{\mathbf{3}}}$

These properties can also be used the other way round, to express two radicals in terms of one radical, as long as the index of the roots is the same. Take the expression $\sqrt{6}·\sqrt{6}$. This can be rewritten as$\sqrt{6·6}=\sqrt{36}=6$.

### The trick to simplify radicals in the denominator

In the previous case, the denominator was a perfect square. When this is not the case, it is very important to use this trick to get rid of radicals in the denominator of fractions:

Multiply both the numerator and the denominator by the radical appearing in the denominator.

Simplify the following radical: $\frac{2}{\sqrt{3}}$

Solution:

Rule 3 is violated since there is a radical in the denominator. This cannot be simplified directly, since 3 is not a perfect square. This let's multiply both the numerator and the denominator by $\sqrt{3}$:

$\frac{2}{\sqrt{3}}=\frac{2}{\sqrt{3}}·\frac{\sqrt{3}}{\sqrt{3}}=\frac{2\sqrt{3}}{\sqrt{3}·\sqrt{3}}=\frac{2\sqrt{3}}{\sqrt{3·3}}=\frac{2\sqrt{3}}{\sqrt{9}}=\frac{2\sqrt{3}}{3}$

Magic! The radical in the denominator vanished, mission completed.

Let's see all of these in action in another couple of examples:

Simplify the following radical: $\sqrt[3]{\frac{24}{125}}$

Solution:

Using the quotient property, we can split this radical into a fraction with two cube roots

$\sqrt[3]{\frac{24}{125}}=\frac{\sqrt[3]{24}}{\sqrt[3]{125}}$

Now, the denominator of this fraction is equal to 5, since${5}^{3}=125$.

What about the numerator? To simplify it, use the product property to split it and isolate a perfect cube:

$\frac{\sqrt[3]{24}}{\sqrt[3]{125}}=\frac{\sqrt[3]{8}·\sqrt[3]{3}}{5}$

We know that $\sqrt[3]{8}=2$, so the final result is

$\sqrt[3]{\frac{24}{125}}=\frac{2\sqrt[3]{3}}{5}$

Are the following radicals fully simplified? If not, simplify them further.

a) $\sqrt[3]{30}$

b) $\sqrt{8}$

c)$\frac{1}{\sqrt{3}}$

d)$\sqrt{\frac{9}{16}}$

Solutions:

a) The radical obeys all three rules and cannot be simplified further. It is fully simplified.

b) A perfect square is present in the radicand as$8=2·4$, and 4 is a perfect square, so this must be simplified further: $\sqrt{8}=\sqrt{4}·\sqrt{2}=2\sqrt{2}$.

c) There is a radicand in the denominator of a fraction, so this expression is not fully simplified. To get rid of this radicand, we must multiply the top and bottom of the fraction by the same radicand, $\sqrt{3}$. This gives $\frac{1·\sqrt{3}}{\sqrt{3}·\sqrt{3}}=\frac{\sqrt{3}}{3}$. This expression is now fully simplified.

d) There is a fraction inside the radicand, so you must use the quotient property to separate this:

$\sqrt{\frac{9}{16}}=\frac{\sqrt{9}}{\sqrt{16}}=\frac{3}{4}$

### Simplifying radicals with variables and exponents

To simplify radicals with variables and exponents, you can follow these steps:

1. Identify the index of the root, and the exponents of each variable in the radicand.

2. If the exponent of a variable in the radicand is greater than or equal to the index of the root, then divide the exponent by the index, to work out how many times the index goes into the exponent.

3. Using the result of the division from step 2, write the variable (outside the radical) with the quotient of the division as an exponent, and if the remainder is greater than zero, write the variable (inside the radical) with the remainder as an exponent.

4. Repeat steps 2 and 3 for each variable in the radicand.

As the value of variables can be positive, negative or zero, we sometimes need to use absolute value when simplifying radicals with variables. This is the case when the index of the root is even and the exponent of the variable is odd after it comes out of the radical.

x3xyFor example, if we simplify $\sqrt{{x}^{7}y}$, then the result is $\left|{x}^{3}\right|\sqrt{xy}$

To avoid using absolute values, we can assume that all variables are positive.

Let's see this more clearly with a few examples, assuming that all variables are positive.

a) Simplify $\sqrt[3]{{x}^{3}{y}^{15}{z}^{20}}$

Solution:

Step 1: In this example, we have a cubic root, as the index is 3. We have 3 variables x , y and z, and their corresponding exponents are 3, 15 and 20.

Steps 2 and 3:

• The exponent of the variable x is 3, which is equal to the index 3. Therefore, we can divide the exponent by the index.

$3÷3=1,witharemainderof0$

We can write x outside of the radical with an exponent of 1, which is the same as writing x, and nothing inside the radical $⇒x\sqrt[3]{}$

• The exponent of the variable y is 15, which is greater than the index 3. Therefore, we can divide the exponent by the index.

$15÷3=5,witharemainderof0$

We can write y outside of the radical with an exponent of 5, and nothing inside the radical $⇒x{y}^{5}\sqrt[3]{}$

• The exponent of the variable z is 20, which is greater than the index 3. Therefore, we can divide the exponent by the index.

$20÷3=6,witharemainderof2$

We can write z outside of the radical with an exponent of 6, and z with an exponent of 2 inside the radical $⇒x{y}^{5}{z}^{6}\sqrt[3]{{z}^{2}}$

So, the radical fully simplified is:

$\overline{)\sqrt[\mathbf{3}]{{\mathbf{x}}^{\mathbf{3}}{\mathbf{y}}^{\mathbf{15}}{\mathbf{z}}^{\mathbf{20}}}\mathbf{=}\mathbf{x}{\mathbf{y}}^{\mathbf{5}}{\mathbf{z}}^{\mathbf{6}}\mathbf{}\sqrt[\mathbf{3}]{{\mathbf{z}}^{\mathbf{2}}}}$

b) Simplify $\sqrt{18{x}^{7}{y}^{8}z}$

Solution:

Step 1: In this example, we have a square root, as the index is 2. We have 3 variables x , y and z, and their corresponding exponents are 7, 8 and 14.

We also have the number 18 inside the radical. Isolating perfect squares, we can say that $18=9·2$, so we can write $\sqrt{18}$ as $\sqrt{9·2}=\sqrt{9}·\sqrt{2}=3\sqrt{2}$. Therefore, we can write 3 outside the radical and 2 stays inside. Now let's focus on the variables.

Steps 2 and 3:

• The exponent of the variable x is 7, which is greater than the index 2. Therefore, we can divide the exponent by the index.

$7÷2=3,withareminderof1$

We can write x outside of the radical with an exponent of 3, and x with an exponent of 1 inside the radical$⇒3{x}^{3}\sqrt{2x}$

• The exponent of the variable y is 8, which is greater than the index 2. Therefore, we can divide the exponent by the index.

$8÷2=4,withareminderof0$

We can write y outside of the radical with an exponent of 4, and nothing inside the radical $⇒3{x}^{3}{y}^{4}\sqrt{2x}$

• The exponent of the variable z is 1, which is less than the index 2. Therefore, we leave it inside the radical $⇒3{x}^{3}{y}^{4}\sqrt{2xz}$

Our final answer in this case is:

$\overline{)\sqrt{\mathbf{18}{\mathbf{x}}^{\mathbf{7}}{\mathbf{y}}^{\mathbf{8}}\mathbf{z}}\mathbf{=}\mathbf{3}{\mathbf{x}}^{\mathbf{3}}{\mathbf{y}}^{\mathbf{4}}\mathbf{}\sqrt{\mathbf{2}\mathbf{x}\mathbf{z}}}$

As we are assuming that all variables are positive, we don't need to use absolute value.

If the radical contains a fraction with variables and exponents in the radicand, then you need to use the same-base division property of exponents $\left({x}^{a}÷{x}^{b}={x}^{a-b}\right)$ to simplify variables with the same base in the numerator and denominator of the fraction.

Simplify $\sqrt[3]{\frac{8{x}^{10}{y}^{5}}{{x}^{3}{y}^{2}}}$

Solution:

Step 1: In this example, we have a cubic root, as the index is 3. We have the number 8 inside the cube root, but we know that $\sqrt[3]{8}=2$, so we can write 2 outside of the radical $⇒2\sqrt[3]{}$

We also have 2 variables x and y. However, both variables appear in the numerator and denominator of the fraction.

Using the same-base division property, we get the following:

$2\sqrt[3]{{x}^{10-3}{y}^{5-2}}=2\sqrt[3]{{x}^{7}{y}^{3}}$

Now we can focus on simplifying the variables.

Steps 2 and 3:

• The exponent of the variable x is 7, which is greater than the index 3. Therefore, we can divide the exponent by the index.

$7÷3=2,witharemainderof1$

We can write x outside of the radical with an exponent of 3, and x with an exponent of 1 inside the radical

$⇒2{x}^{2}\sqrt[3]{x}$

• The exponent of the variable y is 3, which is equal to the index 3. Therefore, we can divide the exponent by the index.

$3÷3=1,witharemainderof0$

We can write y outside of the radical with an exponent of 1, or just y, and nothing inside the radical $⇒2{x}^{2}y\sqrt[3]{x}$

$\overline{)\sqrt[\mathbf{3}]{\frac{\mathbf{8}{\mathbf{x}}^{\mathbf{10}}{\mathbf{y}}^{\mathbf{5}}}{{\mathbf{x}}^{\mathbf{3}}{\mathbf{y}}^{\mathbf{2}}}}\mathbf{=}\mathbf{2}{\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{}\sqrt[\mathbf{3}]{\mathbf{x}}}$

So far, we have shown you how to simplify radicals with positive radicands, but what if the radicand is negative? The way to proceed in this case depends on whether the index of the root is even or odd. Let's see each case in more detail with some examples.

• If the index n of the root is even (i.e. 2, 4, 6, ...), and the radicand is negative, then there is no real solution. Imaginary numbers are required in this case. Only positive radicands can be simplified using the steps already mentioned in this article when the index of the root is even.

$\sqrt{-25}\ne -5$ No real solution!

To be able to simplify radicals with even index and negative radicand, we need to use imaginary numbers. To do this, we say that $\sqrt{-1}=i$.

As a general rule, we have the following:

$\sqrt{-a}=\sqrt{-1·a}=\sqrt{-1}·\sqrt{a}=i·\sqrt{a}=i\sqrt{a}$

You can also write the result the other way around, $i\sqrt{a}=\sqrt{a}i$.

a) Simplify $\sqrt{-25}$

$\sqrt{-25}=\sqrt{-1·25}\phantom{\rule{0ex}{0ex}}=\sqrt{-1}·\sqrt{25}\phantom{\rule{0ex}{0ex}}=i·5\phantom{\rule{0ex}{0ex}}\overline{)\sqrt{\mathbf{-}\mathbf{25}}\mathbf{=}\mathbf{5}\mathbf{i}\mathbf{}}$

b) Simplify $\sqrt{-50}$

$\sqrt{-50}=\sqrt{-1·50}\phantom{\rule{0ex}{0ex}}=\sqrt{-1}·\sqrt{50}$

In this case, we have to use the product property to isolate perfect squares. Note that,$50=25·2$, so $\sqrt{50}$can be rewritten as $\sqrt{25·2}$. We also substitute $\sqrt{-1}$ with $i.$

$=i·\sqrt{25·2}\phantom{\rule{0ex}{0ex}}=i·\sqrt{25}·\sqrt{2}\phantom{\rule{0ex}{0ex}}=i·5·\sqrt{2}\phantom{\rule{0ex}{0ex}}\overline{)\sqrt{\mathbf{-}\mathbf{50}}\mathbf{=}\mathbf{5}\mathbf{i}\sqrt{\mathbf{2}}\mathbf{=}\mathbf{5}\sqrt{\mathbf{2}}\mathbf{i}\mathbf{}}\phantom{\rule{0ex}{0ex}}$
• If the index n of the root is odd (i.e. 3, 5, 7, ...), and the radicand is negative, then there is a real solution. Odd roots of negative radicands can be simplified using real numbers.

a) Simplify $\sqrt[3]{-8}$

Solution:

$\overline{)\sqrt[\mathbf{3}]{\mathbf{-}\mathbf{8}}\mathbf{=}\mathbf{-}\mathbf{2}}$

Since ${\left(-2\right)}^{3}=\left(-2\right)·\left(-2\right)·\left(-2\right)=-8$

When you multiply a negative number by itself three times, the result is also negative.

Therefore, you can in fact simplify the cube root of negative numbers using real numbers!

b) Simplify $\sqrt[3]{-27}$

Solution:

$\overline{)\sqrt[\mathbf{3}]{\mathbf{-}\mathbf{27}}\mathbf{=}\mathbf{-}\mathbf{3}}$

Since ${\left(-3\right)}^{3}=\left(-3\right)·\left(-3\right)·\left(-3\right)=-27$

## Simplifying Radicals - Key takeaways

• A radical is a mathematical expression containing a root in the form $\sqrt[n]{x}$.
• Simplifying radicals means rewriting them in the most simple and fundamental possible way.
• Radicals can be simplified using the product or quotient properties.
• The following three rules must hold for a radical to be fully simplified: 1) No perfect squares in the radicand, 2) No fractions in the radicand, and 3) No radicals in the denominator of a fraction.
• To get rid of radicals in the denominator of fractions: Multiply both the numerator and the denominator by the radical appearing in the denominator.
• When simplifying radicals with variables and exponents, assume that all variables are positive to avoid using absolute values.
• When simplifying negative radicals, if the index of the root is even there is no real solution. Imaginary numbers are required in this case. If the index of the root is odd, negative radicands can be simplified using real numbers.

#### Flashcards in Simplifying Radicals 1

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Radicals can be simplified using the product or quotient properties.

How to simplify radical expressions with fractions?

The quotient rule can be used to split the fraction into two separate radicals.

What are the steps to simplify radicals?

Use the product or quotient properties, and then check that

• There are no perfect squares (except for 1) in the radicand
• There are no fractions in the radicand
• There are no radicals in the denominator of a fraction

How to simplify radicals with variables?

As the value of variables can be positive, negative or zero, we sometimes need to use absolute value when simplifying radicals with variables. This is the case when the index of the root is even and the exponent of the variable is odd after it comes out of the radical.

To avoid using absolute value, we can assume that all variables are positive.

How to simplify radicals with variables and exponents?

To simplify radicals with variables and exponents, you can follow these steps:

• Identify the index of the root, and the exponents of each variable in the radicand.
• If the exponent of a variable in the radicand is greater than or equal to the index of the root, then divide the exponent by the index, to work out how many times does the index go into the exponent.
• Using the result of the division from step 2, write the variable (outside the radical) with the quotient of the division as exponent, and if the remainder greater than zero, write the variable (inside the radical) with the remainder as exponent.
• Repeat the steps 2 and 3 for each variable in the radicand.

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