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Surds: Multiplying and Dividing
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Surds: Multiplying and Dividing

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Surds: Multiplying and Dividing

We can multiply and divide surds together, to form new surds.

We can calculate with surds, subject to a few rules.

1

To multiply, multiply the numbers in the root

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

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So:

3×6=3×6=18\sqrt3 \times \sqrt6 = \sqrt{3 \times 6} =\sqrt{18}

What is 2×5\sqrt{2} \times \sqrt{5}?

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What is 12×5\sqrt{12} \times \sqrt{5} as a single square root?

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1

To divide, divide the numbers in the root

ab=ab\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}

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Therefore:

62=62=3\dfrac{\sqrt6}{\sqrt2} = \sqrt{\dfrac{6}{2}} = \sqrt3

What is 4515\dfrac{\sqrt{45}}{\sqrt{15}} as a single square root?

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We can also multiply and divide mixed surds, like those below, by operating on the number and surd parts separately.

35×763\sqrt{5} \times 7\sqrt{6}

Let's find:

35×763\sqrt{5} \times 7\sqrt{6}

1

Multiply the numbers together: 3×73 \times 7

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2

Now multiply the surds 5×6\sqrt{5} \times \sqrt{6}

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3

Put these two parts together

The final answer is 213021\sqrt{30}. This can't be simplified, as 30 does not have a factor that is a square number.

What is 2410÷8524\sqrt{10} \div 8\sqrt{5}?

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Find 35×433\sqrt{5} \times 4\sqrt{3}

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Let's try a division:

164÷4216\sqrt{4} \div 4\sqrt{2}

1

Treat is as two separate divisions

16÷416\div4 and 4÷24\div2.

2

Find 16÷416\div4

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3

Find 4÷24 \div 2

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4

Put these two parts together

The final answer is 424\sqrt{2}.