Revision:Indices and Surds

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

Laws of Indices

 a^m \times a^n = a^

 a^m \div a^n = a^

 (a^m)^n = a^

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

 a^{-m} = \frac

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

 a^0 = 1

Examples

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

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

 \frac = 5^

Surds

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

There are some basic rules when dealing with surds

 \sqrt + \sqrt = 2\sqrt

 6\sqrt - 2\sqrt = 4\sqrt

 \sqrt \times \sqrt = \sqrt

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

Also notice the special case

 \sqrt \times \sqrt = a^{\frac} \times a^{\frac} = 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{\sqrt}

To rationalise this multiply by effectively 1

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

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

For a more complex term

 \frac}}

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

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

So to get rid of the denominator surd we multiply  \frac}} by  \frac{}}} like so.

 \frac}} \times \frac{}}}

 = \frac{(1+\sqrt)(1+\sqrt)}{(1-\sqrt)(1+\sqrt)}

 = \frac + \sqrt + \sqrt}

 = -\frac - \frac{\sqrt5} - \frac{\sqrt2} - \frac{\sqrt10}

In general

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