It follows logically that if we wanted to multiply two surds together we could combine them back in the same way. That is \sqrt{a} \times \sqrt{b}=\sqrt{a \times b}\,.

If we want to find the product of more complicated surds, that may have a term outside the surd, we can use the properties of multiplication to rearrange the expression to make it easier.

Examples

Example 1

Express \sqrt{19} \times \sqrt{17} as a single surd. That is, in the form \sqrt{n}.

Worked Solution

Create a strategy

Use the rule \sqrt{A} \times \sqrt{B}=\sqrt{A \times B}.

Apply the idea

\displaystyle \sqrt{19} \times \sqrt{17}	\displaystyle =	\displaystyle \sqrt{19 \times 17}	Apply the rule
	\displaystyle =	\displaystyle \sqrt{323}	Evaluate 19 \times 17

Example 2

Simplify: 4\sqrt{33} \times 4\sqrt{11}

Worked Solution

Create a strategy

Multiply the integer terms together and the surds together.

Apply the idea

\displaystyle 4\sqrt{33} \times 4\sqrt{11}	\displaystyle =	\displaystyle 4 \times 4 \times \sqrt{33 \times 11}	Rearrange the terms
	\displaystyle =	\displaystyle 16 \sqrt{363}	Perform the multiplication

We can simplify our answer since 363 = 121 \times 3, where 121 is a perfect square.

\displaystyle 16 \sqrt{363}	\displaystyle =	\displaystyle 16 \times \sqrt{121} \times \sqrt{3}	Write the surd as a product of its factors
	\displaystyle =	\displaystyle 16 \times 11 \times \sqrt{3}	Evaluate \sqrt{121}
	\displaystyle =	\displaystyle 176 \sqrt{3}	Simplify

Example 3

Expand and simplify: 7\sqrt{2} \left(\sqrt{7}+6 \right)

Worked Solution

Create a strategy

Use the distributive law: A(B-C)=AB-AC

Apply the idea

\displaystyle 7\sqrt{2} \left(\sqrt{7}+6 \right)	\displaystyle =	\displaystyle 7\sqrt{2} \times \sqrt{7} + 7\sqrt{2} \times 6	Apply the distributive law
	\displaystyle =	\displaystyle 7\sqrt{2 \times 7} + (7\times 6)\sqrt{2}	Multiply the surds and integers
	\displaystyle =	\displaystyle 7\sqrt{14} + 42\sqrt{2}	Evaluate the products

Idea summary

To multiply terms involving surds use the rule:

a\sqrt{m} \times b\sqrt{n}=ab \times \sqrt{mn}

Divide surds

It follows that we can treat the division of surds in the same way. That is \sqrt{a} \div \sqrt{b}=\sqrt{a \div b}. We can also split terms up so that we can evaluate the quotient of the integer terms, and the surd terms separately.

Examples

Example 4

Simplify: \dfrac{\sqrt{6}}{\sqrt{2}}

Worked Solution

Create a strategy

Use the rule: \dfrac{\sqrt{A}}{\sqrt{B}}=\sqrt{\dfrac{A}{B}}

Apply the idea

\displaystyle \dfrac{\sqrt{6}}{\sqrt{2}}	\displaystyle =	\displaystyle \sqrt{\dfrac{6}{2}}	Apply the rule
	\displaystyle =	\displaystyle \sqrt{3}	Perform the division

Example 5

Simplify \dfrac{21\sqrt{80}}{3\sqrt{5}}.

Worked Solution

Create a strategy

Split the surd parts and integer parts.

Apply the idea

\displaystyle \dfrac{21\sqrt{80}}{3\sqrt{5}}	\displaystyle =	\displaystyle \dfrac{21}{3} \times \dfrac{\sqrt{80}}{\sqrt{5}}	Split the integer and surd parts
	\displaystyle =	\displaystyle \dfrac{21}{3} \times \sqrt{\dfrac{80}{5}}	Write the surds as one surd
	\displaystyle =	\displaystyle 7\sqrt{16}	Evaluate the divisions
	\displaystyle =	\displaystyle 7\times 4	Evaluate the square root
	\displaystyle =	\displaystyle 28	Evaluate

Reflect and check

Similar to the multiplication example above, we can approach this question by first simplifying \sqrt{80}=\sqrt{16\times 5}=4\sqrt{5}.

We can rewrite the original expression:

\displaystyle \dfrac{21\sqrt{80}}{3\sqrt{5}}	\displaystyle =	\displaystyle \dfrac{21 \times 4\sqrt{5}}{3\sqrt{5}}	Simplify \sqrt{80}
	\displaystyle =	\displaystyle \dfrac{84}{3} \times \dfrac{\sqrt{5}}{\sqrt{5}}	Separate the integer and surd parts
	\displaystyle =	\displaystyle 28	Simplify

Idea summary

To divide terms involving surds use the rule:

a\sqrt{m} \div b\sqrt{n}=\dfrac{a}{b} \times \sqrt{\dfrac{m}{n}}

Outcomes

VCMNA355 (10a)

Define rational and irrational numbers and perform operations with surds and fractional indices.

2.03 Multiplying and dividing surds

Ideas

Multiply surds

Examples

Example 1

Example 2

Example 3

Divide surds

Examples

Example 4

Example 5

Outcomes

VCMNA355 (10a)

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