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Surds: Basics

Surds are irrational roots, which can be multiplied and divided using certain rules.

A surd is a root of an integer that is irrational.

$\dfrac{1}{2}$ , $17$ , $\dfrac{45}{7}$ and $0.34$ are all examples of rational numbers because they could be written as fractions.

Is $\dfrac{9}{10}$ rational?

$\sqrt{9} = 3$ is a rational answer, as it gives an integer.

We cannot write $\sqrt{10}$ as a rational number, so we call it a surd.

$\sqrt2$ , $\pi$ and $\sqrt{30}$ are all examples of surds as they cannot be written as exact fractions.

Which of the following is a surd?

We can simplify surds by splitting into two factors. One should be a perfect square number, and the other a surd.

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

We can understand this better using indices.

$\sqrt{ab} = (ab)^{\frac{1}{2}} = a^{\frac{1}{2}} \times b^{\frac{1}{2}} \rightarrow \sqrt{a} \times \sqrt{b}$

Let's simplify $\sqrt{18}$

Think of two numbers that multiply to make $18$

We have $1 \times 18$ or $2 \times 9$ or $3 \times 6$ .

Which multiplication fact above contains a square number?

So, we can split $\sqrt{18}$ into two different roots

$\sqrt{18} = \sqrt{2 \times 9} = \sqrt{2} \times \sqrt{9}$

What is $\sqrt{9}$ ?

We now have $\sqrt{2} \times 3$

This can be written as one mixed surd: $3\sqrt{2}$

This is the simplest version

$\sqrt{18} = 3\sqrt{2}$

Nice! 👍

Express $\sqrt{72}$ in the form $a\sqrt{2}$ .

Think of factors that multiply together to make $72$ .

Hang on - this question gives us a hint, one of the factors of $72$ must be $2$ as it's in the final form!

If $2$ is one factor of $72$ , what must the other factor be?

This leaves us with two roots

We have: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}$

Simplify $\sqrt{36}$

Finally, express $\sqrt36\sqrt2$ in the form $a\sqrt{2}$

This is the simplest form!

$\sqrt{72} = 6\sqrt{2}$

Great work! 😃

Simplify $\sqrt{63}$

Key rule: to divide, divide the numbers in the root

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

So:

$\sqrt2 \times \sqrt5 = \sqrt{2 \times 5} =\sqrt{10}$

As with normal numbers, we are able to calculate with surds too. There are a couple of important rules we will need to learn when dealing with surds.

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

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

Key rule: to multiply, multiply the numbers in the root

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

Which of the following is not a surd?