2 Making sense of algebra
Dealing with non-unit fractions A non-unit fraction has a numerator (the number on top) that is not 1. 5 2 For example, and are non-unit 7 3 fractions.
2
3
Sometimes you may have to work with indices that are non-unit fractions. For example x 3 or y 4 . To find the rule for working with these, you have to think back to the law of indices for raising a power to another power. Look at these examples carefully to see how this works: x3
2
(x 3 )2
3
( y 4 )3
2 1 × 2 is 3 3
1
1 4
1
y4
×3 =
3 4
You already know that a unit-fraction gives a root. So we can rewrite these expressions using root signs like this: 1
1
( )2
( 3 )2 and ( y )3
( y )3
3
2
3 So, ( ) ( 3 )2 and ( y ) ( y ) .
It is possible that you would want to reverse the order of calculations here and the result will be the m same. x n ( n x )m = n x m , but the former tends to work best.
x m× n = (x n )m
m
1
In general terms: x n
1
( n x )m
Worked example 19 Work out the value of: 2
(a) 27 3
(b) 25 2
(a)
27 3 = ( 3 27 )2 = (3) =9
2
REWIND
(b)
25
You saw in chapter 1 that a ‘vulgar’ a fraction is in the form . W b
15
3
= 25 2 = ( 25 )
3
15
2 1 = 2 × so you square the cube root of 27. 3 3
Change the decimal to a vulgar fraction. 3 = 3 × 1, so you need 2 2 to cube the square root of 25.
= (5)2 = 125
Sometimes you are asked to find the value of the power that produces a given result. You have already learned that another word for power is exponent. An equation that requires you to find the exponent is called an exponential equation.
Worked example 20 If 2x = 128 find the value of x. 2 x = 128 2 = 128 7
∴x =7
40
Unit 1: Algebra
Remember this means 2 = x 128 . Find the value of x by trial and improvement.