Mathematics

1. INDICES
PART 2

2. CONTENTS
• Laws of exponents
• Negative and zero indices law
• Fractional indices
• Challenging questions

3. LAWS OF EXPONENTS
 𝑥1 = 𝑥
 𝑥0
= 1
 𝑥𝑚 × 𝑥𝑛 = 𝑥𝑚+𝑛
 𝑥𝑚 ÷ 𝑥𝑛 = 𝑥𝑚−𝑛
 𝑥𝑚 𝑛 = 𝑥𝑚×𝑛

4. NEGATIVE INDICES
Powers don’t have to be always positive whole numbers.
Simplify: 𝑑4 ÷ 𝑑5 =?
Using the law of indices: 𝑑4 ÷ 𝑑5 = 𝑑4−5 = 𝑑−1. This is an example of negative index.
Using the law of arithmetic: 𝑑4 ÷ 𝑑5 =
𝑑×𝑑×𝑑×𝑑
𝑑×𝑑×𝑑×𝑑×𝑑
=
1
𝑑
𝑑−1
=
1
𝑑
A negative power is often referred to as a reciprocal (𝑥−𝑚 =
1
𝑥𝑚 is the reciprocal of 𝑥𝑚)

5. SOLVE
• 𝑝−2
• 2−3
•
1
9
• 52 ÷ 55
•
1
105

6. ZERO INDICES
Simplify: 𝑗2 ÷ 𝑗2 =?
By law of indices:𝑗2
÷ 𝑗2
= 𝑗2−2
= 𝑗0
By law of arithmetic:𝑗2 ÷ 𝑗2 = 1
𝑗0 = 1
P.S. This rule is true for all non-zero values of 𝑗.

7. SOLVE
1. (−5)0
2. 𝑎100
÷ 𝑎100
3. (100)−5× (100)5

8. FRACTIONAL INDICES
𝑥
1
𝑛
𝒏 – shows the root of the number
For example: 𝑎
1
3 = 3
𝑎

9. SOLVE
1. 32
1
5
2.
3
64
3. (81
1
4)2

10. CHALLENGING QUESTIONS
1. Work out 4 × 108
× 5 × 10−6
Give your answer in standard form.
2. A group of friends write these cards.
Anna Joe Julia Ben Bella
Simplify each expression.
Arrange the friends so the values of their expressions are in ascending order if
i. 𝒙 = 𝟓 ii. 𝒙 =
𝟏
𝟓
𝒙𝟓
× 𝒙−𝟒
𝒙
𝟏
𝟖
𝟏𝟔 𝒙𝟕
𝒙𝟖
𝒙−𝟏𝟎
× 𝒙𝟏𝟎
𝒙−𝟒
÷ 𝒙−𝟔

11. THANKS FOR ATTENTION