Introduction

To multiply square roots, we simply multiply the numbers under the root. This is called the product rule for radicals.

To demonstrate, let's simplify the expression \sqrt{2} \cdot \sqrt{5} by applying the product rule:

\sqrt{2} \cdot \sqrt{5} = \sqrt{2 \cdot 5} = \sqrt{10}

The product rule also works for cube roots. For example,

\sqrt[3]{2} \cdot \sqrt[3]{5} = \sqrt[3]{2 \cdot 5} = \sqrt[3]{10}.

In general, the product rule works for any product of n th roots. So for any n, we have

\sqrt[n]{2} \cdot \sqrt[n]{5} = \sqrt[n]{2 \cdot 5} = \sqrt[n]{10}.

Watch out! We cannot use the product rule when we have the product of different radicals, such as square roots and cube roots. For example, the expression \sqrt{2} \times \sqrt[3]{5} cannot be simplified because the radicals \sqrt{\phantom{x}} and \sqrt[3]{\phantom{x}} are different.

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Simplifying Square Roots

The product rule can often be used to simplify square roots.

For example, consider \sqrt{18}. Note that 18=9 \cdot 2, where 9 is a perfect square. So, we have

\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2}

Alternatively, if we're given a simplified square root and we want to convert it to the square root of a single number then we can work backwards using the product rule:

3 \sqrt{2} = \sqrt{9} \cdot \sqrt{2} = \sqrt{9 \cdot 2} = \sqrt{18}

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Example: Separating a Square Root Into a Product of Two Square Roots

Simplify \sqrt{12}.

EXPLANATION

Note that 12=4 \cdot 3, where 4 is a perfect square. So, applying the product rule, we have

\sqrt{12} =\sqrt{4 \cdot 3}= \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}.

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Practice: Separating a Square Root Into a Product of Two Square Roots

$\sqrt{50} =$

a
 $20\sqrt{5}$
b
 $25\sqrt{2}$
c
 $2\sqrt{5}$
d
 $10$
e
 $5\sqrt{2}$
Practice: Separating a Square Root Into a Product of Two Square Roots

$\sqrt{21} =$

a
 $9\cdot\sqrt{7}$
b
 $3\sqrt{7}$
c
 $49\sqrt{2}$
d
 $7\sqrt{2}$
e
 $\sqrt{3}\cdot\sqrt{7}$
Example: Writing a Product of Two Square Roots as a Single Square Root

Write 3 \sqrt{5} as the square root of a single number.

EXPLANATION

Note that 3 = \sqrt{9}. So, applying the product rule, we have

3 \sqrt{5} = \sqrt{9}\cdot\sqrt{5}= \sqrt{9\cdot 5} = \sqrt{45}.

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Practice: Writing a Product of Two Square Roots as a Single Square Root

$2\sqrt{7}= $

a
 $\sqrt{12}$
b
 $\sqrt{28}$
c
 $\sqrt{49}$
d
 $\sqrt{14}$
e
 $\sqrt{26}$
Practice: Writing a Product of Two Square Roots as a Single Square Root

$\sqrt{3}\cdot\sqrt{2}= $

a
 $\sqrt{6}$
b
 $\sqrt{5}$
c
 $6$
d
 $3\sqrt{2}$
e
 $2\sqrt{3}$
The Product Rule With Cube Roots

The product rule applies to cube roots as well.

For example, consider \sqrt[3]{24}. Note that 24=8 \cdot 3, where 8 is a perfect cube. So, we have

\sqrt[3]{24} = \sqrt[3]{8 \cdot 3} = \sqrt[3]{8} \cdot \sqrt[3]{3} = 2 \sqrt[3]{3}.

Again, if we're given a simplified cube root and we want to convert it to the cube root of a single number then we can work backwards using the product rule:

2 \sqrt[3]{3} = \sqrt[3]{8} \cdot \sqrt[3]{3} = \sqrt[3]{8 \cdot 3} = \sqrt[3]{24}

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Example: The Product Rule With Cube Roots

Express \sqrt[3]{2}\cdot\sqrt[3]{3} as the cube root of a single number.

EXPLANATION

Using the product rule, we have

\sqrt[3]{2}\cdot\sqrt[3]{3}= \sqrt[3]{2\cdot 3} = \sqrt[3]{6}.

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Practice: The Product Rule With Cube Roots

$\sqrt[3]{24} =$

a
 $2\sqrt[3]{3}$
b
 $4\sqrt[3]{3}$
c
 $6$
d
 $3\sqrt[3]{2}$
e
 $8\sqrt[3]{3}$
Practice: The Product Rule With Cube Roots

$\sqrt[3]{3}\cdot\sqrt[3]{4}= $

a
 $3$
b
 $ 2\sqrt[3]{3}$
c
 $ \sqrt[3]{12}$
d
 $ \sqrt[3]{24}$
e
 $ \sqrt[3]{6}$
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