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Revision:Indices and Surds

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Indices and Surds

Indices are used to describe the general term for in say x^2 . There are a few laws to know when manipulating expressions involving indices.

1 Laws of Indices
- 1.1 Examples
2 Surds
- 2.1 Difference of Two Squares
- 2.2 Rationalising Surds

Laws of Indices

$a^m \times a^n = a^{m+n}$

$a^m \div a^n = a^{m-n}$

$(a^m)^n = a^{mn}$

$a^{\frac{1}{m}} = \sqrt[m]{a}$

$a^{-m} = \frac{1}{a^m}$

$a^{\frac{m}{n}} = \sqrt[n]a^m$

a^0 = 1

Examples

$\sqrt[5]4^3 = (4^3)^{\frac{1}{5}} = (4^{\frac{3}{5}})$

$5^3 \times 5^4 = 5^7$

$\frac{5^5}{5^2} = 5^{3}$

Surds

Surds are basically an expression involving a root, squared or cubed etc...

There are some basic rules when dealing with surds

$\sqrt{a} + \sqrt{a} = 2\sqrt{a}$

$6\sqrt{a} - 2\sqrt{a} = 4\sqrt{a}$

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

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

Also notice the special case

$\sqrt{a} \times \sqrt{a} = a^{\frac{1}{2}} \times a^{\frac{1}{2}} = a$

Difference of Two Squares

x^2 - y^2 = (x + y)(x - y) This is called the difference of two squares

Rationalising Surds

When you have a fraction where both the nominator and denominator are surds, rationalising the surd is the process of getting rid of the surd on the denominator.

To rationalise a surd you multiply top and bottom by fraction that equals one. Take the example shown below

$\frac{1}{\sqrt{2}}$

To rationalise this multiply by effectively 1

$\frac{1}{\sqrt2} \times \frac{\sqrt 2}{\sqrt{2}}$

Can you see why $\frac{\sqrt2}{\sqrt{2}}$ was chosen? This is because $\sqrt{2} \times \sqrt{2} = 2$ so the denominator becomes surd free.

For a more complex term

$\frac{1+\sqrt{2}}{1 - \sqrt{5}}$

First of all, we need to get rid of the surd expression on the bottom, you should remember the difference of two squares formula.

a^2 - b^2 = (a + b)(a - b)

suppose a = 1 and b = $\sqrt{5}$

$1^2 - 5 = (1+\sqrt{5})(1 - \sqrt{5}) = -4$

So to get rid of the denominator surd we multiply $\frac{1+\sqrt{2}}{1 - \sqrt{5}}$ by $\frac{{1+\sqrt{5}}}{1+\sqrt{5}}$ like so.

$\frac{1+\sqrt{2}}{1 - \sqrt{5}} \times \frac{{1+\sqrt{5}}}{1+\sqrt{5}}$

$= \frac{(1+\sqrt{2})(1+\sqrt{5})}{(1-\sqrt{5})(1+\sqrt{5})}$

$= \frac{1 + \sqrt{5} + \sqrt{2} + \sqrt{10}}{1^2 - 5}$

$= -\frac{1}{4} - \frac{\sqrt5}{4} - \frac{\sqrt2}{4} - \frac{\sqrt10}{4}$

In general

Fractions in the form $\sqrt{\frac{1}{a}}$ multiply top and bottom by $\sqrt{a}$
Fractions in the form $\frac{1}{a + \sqrt{b}}$ multiply the top and bottom by $a - \sqrt{b}$
Fractions in the form $\frac{1}{a - \sqrt{b}}$ multiply the top and bottom by $a + \sqrt{b}$