Negative indices It is possible to express terms in negative powers. This needs to be done following certain rules. Can you write an equivalent expression for the term x–n? x × x 1 x × x = 1 x2 Look at the division: x2 ÷ x4 = = x × x × x × x But using the rule that: xm ÷ xn = x(m – n) x2 ÷ x4 = x(2 – 4) = x–2 Teacher notes Discuss the general form of each result where x is any number and m and n are integers. 1 x2 This means that: x–2 = In general, the rule for expressing terms in negative powers is: x–n = 1 xn 3

Negative indices

Negative indices Terms can be expressed in a variety of ways involving negative indices. Can you write equivalent terms for these expressions? 1 u 2 b4 u–1 = 2b–4 = Teacher notes A number or term raised to the power of –1 is the reciprocal of the number or term. x2 y3 2a (3 – b)2 x2y–3 = 2a(3 – b)–2 = 5

Reciprocals A number raised to the power of –1 gives us the reciprocal of that number. The reciprocal of a number is what we multiply the number by to get 1. a 1 The reciprocal of a is: . b a a b Teacher notes Tell pupils that if a number is written as a fraction, we can easily find the reciprocal by swapping the numerator and the denominator. You could ask pupils to show why a/b × b/a will always equal 1. We can find reciprocals on a calculator using the x–1 key. The reciprocal of is: . What is the reciprocal of the number 9?

Finding the reciprocals Teacher notes Tell pupils that when we find the reciprocal of a decimal we can first write it as a decimal and then invert it. If we had a calculator we could also work out 1 ÷ the number or use the x–1 key. Improper fractions such as 7/3 can be written as mixed numbers if required. If a number is given as a decimal then its reciprocal is usually given as a decimal too.

Match the reciprocal pairs Teacher notes Establish that we have to find pairs of numbers that multiply together to make one. Each will be the reciprocal of the other. Point out that one number in the pair must be less than 1 and one number must be more than 1. Encourage pupils to convert decimals into fractions and mixed numbers into-top heavy fractions. The resulting fraction can then be reversed to find its reciprocal.

Fractional indices It is possible to raise terms to the power of fractional indices. What would be the value of 9½ × 9½? 9½ × 9½ = 9½ + ½ xm × xn = x(m + n) = 91 = 9 1 3 1 3 1 3 What would be the value of 8 × 8 × 8 ? Teacher notes Elicit from pupils that the square root of a number multiplied by the square root of the same number always equals that number. We could also think of this as square rooting a number and then squaring it. Since squaring and square rooting are inverse operations this takes us back to the original number. The same is true of cube roots. If we cube root a number and then cube it we come back to the original number. 1 3 1 3 1 3 1 3 1 3 1 3 xm × xn = x(m + n) 8 × 8 × 8 = 8 + + = 81 = 8

Fractional indices As shown on the previous slide, indices can also be fractional. Can you write an equivalent expression for the term x½? x × x = 1 2 2 x + = 1 x1 = x But, x × x = x The square root of x. So, x = x 1 2 Similarly, Teacher notes Discuss the fact that when we square the square root of a number we end up with the original number. Similarly, we get back to where we started when we cube the cube root of a number. 3 1 x × x × x = x + + = 1 3 x1 = x But, x × x × x = 3 x The cube root of x. So, x = x 1 3 3 12

Fractional indices It is possible to express terms in fractional powers. This needs to be done following certain rules. 1 In general, x = x n n We can write x as x . × m m n 1 Using the rule that (xm)n = xmn, we can write: x ×m = 1 n 1 n (x )m = n (x)m n x = 1 or m × n 1 (xm) = n xm x = xm = (x )m m n 13

Fractional indices As shown on the previous slide, fractional indices follow certain rules. What is the value of 25 ? 3 2 We can think of 25 as 25½ × 3. 3 2 Using the rule that (xm)n = xmn, we can write that: 25½ × 3 = (25)3 Teacher notes Explain that the denominator of the power 2 square roots the number and the numerator of the power 3 cubes the number. = (5)3 = 125

Index laws summary

Evaluate the calculations