Here you will learn about dividing radicals, including when radical expressions can be divided, how to simplify expressions using division and how to do the calculations.

Students first learn how to work with radicals in pre-algebra and expand this knowledge as they move into an algebra 1 and algebra 2 class.

Dividing radicals is where radicals are combined using the division rule to be written as a rationalized radical expression. To divide radicals, use the quotient rule:

\sqrt{\cfrac{m}{n}}=\cfrac{\sqrt{m}}{\sqrt{n}}=\sqrt{m}\div\sqrt{n}

Once the radical expression is written in the form of a numerator divided by a denominator, you must determine if they divide evenly.

If the radicands in the numerator and denominator do not divide, you cannot leave the expression with a radical in the denominator. If there is a radical in the denominator, you must simplify by rationalizing it.

Examples

In the second example, you cannot divide the numerator and the denominator of the expression for two reasons.

The first is that both expressions are not radical expressions. In order to divide radicals, both expressions have to be radicals.

The second reason is that 2 is not a factor of 5. So, \cfrac{2}{\sqrt{5}} cannot be divided, but notice that the denominator has a radical expression which means that the expression needs to be rationalized.

In order to rationalize the denominator, multiply the numerator and the denominator by the radical in the denominator as shown above.

Step-by-step guide: Rationalizing the denominator

Remember that surds can be further simplified using the product rule

\sqrt{a\times{b}}=\sqrt{a}\times\sqrt{b}

## Common Core State Standards

How does this relate to high school math?

• Number and Quantity – High School: (HSN-RN.B.3)
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

1. Substitute known values into the quotient rule.
2. Simplify the expression.

### Example 1: simplifying radical expressions

Simplify the expression \cfrac{\sqrt{24}}{\sqrt{8}}.

1. Substitute known values into the quotient rule.

\cfrac{24}{8}=\sqrt{\cfrac{24}{8}}

2Simplify the expression.

\sqrt{\cfrac{24}{8}}=\sqrt{3}.

So \cfrac{\sqrt{24}}{\sqrt{8}} simplified fully is \sqrt{3}.

### Example 2: simplifying radical expressions

Simplify the expression \cfrac{4\sqrt{15}}{2\sqrt{5}} fully.

Substitute known values into the quotient rule.

Simplify the expression.

### Example 3: simplifying radical expressions

Simplify fully the radical expression \cfrac{\sqrt{27}}{3\sqrt{3}}.

Substitute known values into the quotient rule.

Simplify the expression.

### Example 4: rationalize denominator

Simplify the radical expression \cfrac{5\sqrt{2}}{\sqrt{3}} fully.

Substitute known values into the quotient rule.

Simplify the expression.

### Example 5: dividing radical expressions

Simplify the expression \cfrac{\sqrt{40}}{\sqrt{5}}.

Substitute known values into the quotient rule.

Simplify the expression.

### Example 6: dividing radical expressions

Simplify the expression \cfrac{4\sqrt{15}+\sqrt{60}}{\sqrt{5}}.

Substitute known values into the quotient rule.

Simplify the expression.

### Example 7: using a conjugate to rationalize the denominator

Substitute known values into the quotient rule.

Simplify the expression.

### Teaching tips for dividing radicals

• Reinforce perfect square numbers with students.

• Review with students how to break numbers down into their factors.

• Instead of having students practice skills with worksheets, incorporate game playing.

### Easy mistakes to make

• Dividing a radical with a coefficient
For example, in the expression \cfrac{4}{\sqrt{2}} thinking that you can divide.
\cfrac{4}{\sqrt{2}} {2} or \sqrt{2}

In this case, you cannot divide, but you can rationalize the denominator.

\cfrac{4}{\sqrt{2}}\times\cfrac{\sqrt{2}}{\sqrt{2}}=\cfrac{4\sqrt{2}}{\sqrt{4}}=\cfrac{4\sqrt{2}}{2}=2\sqrt{2}

• Forgetting to divide the entire numerator by the denominator
For example, in the expression \cfrac{2\sqrt{15}-\sqrt{10}}{\sqrt{5}} only dividing one part of the numerator by the denominator,

\cfrac{2\sqrt{15}-\sqrt{10}}{\sqrt{5}} {2}\sqrt{3}-\sqrt{10}

In this case, the correct way to divide is to divide \sqrt{5} into both parts of the numerator, 2\sqrt{15}\div\sqrt{5} and \sqrt{10}\div\sqrt{5}.

So, \cfrac{2\sqrt{15}-\sqrt{10}}{\sqrt{5}}=2\sqrt{3}-\sqrt{2}

1. Simplify the expression \cfrac{\sqrt{40}}{\sqrt{4}} completely.

\sqrt{10}

\sqrt{36}

2\sqrt{5}

\sqrt{160}

For the expression \cfrac{\sqrt{40}}{\sqrt{4}}, the radical in the denominator divides into the radical in the numerator because 4 is a factor of 40.

\sqrt{40}\div\sqrt{4}=\sqrt{\cfrac{40}{4}}=\sqrt{10}

2. Simplify the expression \cfrac{\sqrt{90}}{\sqrt{2}} completely.

5 \sqrt{3}

\sqrt{45}

3 \sqrt{5}

\sqrt{88}

For the expression \cfrac{\sqrt{90}}{\sqrt{2}} , the radical in the denominator divides into the radical in the numerator because 2 is a factor of 90.

\sqrt{90} \div \sqrt{2}=\sqrt{45}

\cfrac{\sqrt{90}}{\sqrt{2}}=\sqrt{45}

\sqrt{45} can be simplified further because 9 is the largest square factor of 45.

\begin{aligned}\sqrt{45}&=\sqrt{9}\times\sqrt{5} \\\\ &=3\times\sqrt{5} \\\\ &=3\sqrt{5} \end{aligned}

3. Simplify the expression \cfrac{7 \sqrt{5}}{\sqrt{6}} fully.

7 \sqrt{30}

\cfrac{7 \sqrt{30}}{6}

\cfrac{7 \sqrt{5}}{36}

\cfrac{7 \sqrt{5}}{6}

For the expression \cfrac{7 \sqrt{5}}{\sqrt{6}}, the radical in the denominator does not divide into the radical in the numerator because 6 is not a factor of 5.

So in order to remove the radical from the denominator, you have to rationalize the denominator by multiplying the numerator and the denominator by \sqrt{6}.

\cfrac{7\sqrt{5}}{\sqrt{6}}\times\cfrac{\sqrt{6}}{\sqrt{6}}=\cfrac{7\sqrt{30}}{\sqrt{36}}=\cfrac{7\sqrt{30}}{6}

\cfrac{7\sqrt{30}}{6}

30 does not have any square factors except 1 so this cannot be simplified any further.

4. Simplify the expression \cfrac{12 \sqrt{12}}{2 \sqrt{6}} fully.

\cfrac{12\sqrt{6}}{2}

2\sqrt{6}

\cfrac{6\sqrt{2}}{\sqrt{6}}

6\sqrt{2}

The expression \cfrac{12\sqrt{12}}{2\sqrt{6}} can be divided because the coefficient of the denominator divides the coefficient of the numerator (2 is a factor of 12).

Also, the radical in the denominator divides the radical in the numerator, (6 is a factor of 12).

You can remove the radical from the denominator by dividing.

12\div2=6

\sqrt{12} \div \sqrt{6}=\sqrt{2}

\cfrac{12\sqrt{12}}{2\sqrt{6}}=6\sqrt{2}

5. Divide the expression \cfrac{\sqrt{2}}{3\sqrt{11}}.

\cfrac{\sqrt{22}}{3}

\cfrac{\sqrt{22}}{33}

\sqrt{22}

\cfrac{\sqrt{22}}{11}

For the expression \cfrac{\sqrt{2}}{3 \sqrt{11}} ,  the denominator does not divide evenly into the numerator.

So, to remove the radical from the denominator, you have to rationalize the denominator.

To rationalize the denominator, multiply the denominator and the numerator by \sqrt{11}.

\cfrac{\sqrt{2}}{3\sqrt{11}}\times\cfrac{\sqrt{11}}{\sqrt{11}}=\cfrac{\sqrt{22}}{3\sqrt{121}}=\cfrac{\sqrt{22}}{3\times11}=\cfrac{\sqrt{22}}{33}

6. Simplify the expression \cfrac{4 \sqrt{15}+2 \sqrt{15}}{3 \sqrt{5}} fully.

6\sqrt{3}

6\sqrt{5}

2\sqrt{3}

2\sqrt{5}

For the expression \cfrac{4 \sqrt{15}+2 \sqrt{15}}{3 \sqrt{5}}, the numerator can be simplified because the numbers under the radicals are the same.

So, the radical in the denominator can be removed by division.

So, \cfrac{4\sqrt{15}+2\sqrt{15}}{3\sqrt{5}}=\cfrac{6\sqrt{15}}{3\sqrt{5}}

The numerator and the denominator can divide because 6\div{3}=2 and \sqrt{15}\div\sqrt{5}=\sqrt{3}

So, \cfrac{6\sqrt{15}}{3\sqrt{5}}=2\sqrt{3}

Can exponential expressions be represented by radical expressions?

Yes, exponential expressions can be written as exponential expressions using rational exponents.

Can cube root expressions be simplified?

Yes, cube root expressions can be simplified using the same strategies as simplifying square root expressions.

Can there be polynomials under a radical sign?

Yes, there can be polynomials under radical signs such as \sqrt{x^{2}+2x+1}.

Can there be radicals in rational expressions?

Yes, there can be radicals in rational expressions, such as in the expressions, \cfrac{\sqrt{x}}{\sqrt{2x-4}}.

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