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1.4 Working with indices

1.4 Working with indices

You can write the numbers in the boxes as powers. 9 × 27 = 243 32 × 8 = 256
Look at the indices. 2 + 3 = 5 and 5 + 3 = 8.
3 × 3 = 3 5 2 × 2 = 2 8
5
3
3
2

"is is an example of a general result.
To multiply powers of a number, add the indices. A × A = A m + n
n
m
9 × 9 = 81 ⇒ 3 × 3 = 3 2 + 2 = 4
2
2
4
4 × 8 = 32 ⇒ 2 × 2 = 2 2 + 3 = 5
5
3
2
"e multiplications above can be written as divisions.

243 ÷ 27 = 9 256 ÷ 8 = 32
You can write the numbers as powers.
Again, look at the indices. 5 − 3 = 2 and 8 − 3 = 5. 3 ÷ 3 = 3 2 2 ÷ 2 = 2 5
8
5
3
3

"is shows that:
To divide powers of a number, subtract the indices. A ÷ A = A m – n
m
n
27 ÷ 3 = 9 ⇒ 3 ÷ 3 = 3 3 − 1 = 2
2
3
1
1
4 ÷ 8 = ⇒ 2 ÷ 2 = 2 2 − 3 = −1
−1
3
2
2
Worked example 1.4
a Write each expression as a power of 5. i 5 × 5 ii 5 ÷ 5 3
2
3
2
b Check your answers by writing the numbers
as decimals.
a i 5 × 5 = 5 2 + 3 = 5 2 + 3 = 5
5
3
2
−1 1
ii 5 ÷ 5 = 5 2 − 3 = 5 = 2 − 3 = −1
2
3
5
b i 25 × 125 = 3125 3125 is 5 5
1
ii 25 ÷ 125 = = 0.2
5
) Exercise 1.4
1 Simplify each expression. Write your answers in index form.
a 5 × 5 b 6 × 6 c 10 × 10 d a × a × a e 4 × 4
4
2
2
2
3
2
3
4
5
3
2 Simplify each expression. Leave your answers in index form where appropriate.
a 2 × 2 b 8 × 8 c a × a d 2 × 2 e b × b 4
3
3
3
4
2
2
3
5
3
3 Simplify each expression.
a 3 ÷ 3 b k ÷ k c 10 ÷ 10 d 5 ÷ 5 e 7 ÷ 7 1
4
2
2
6
3
4
5
4
4 Simplify each expression.
a 2 ÷ 2 b 2 ÷ 2 c 2 ÷ 2 d 2 ÷ 2 e 2 ÷ 2 6
4
2
3
2
2
4
2
2
4
12 1 Integers, powers and roots