Free Maths Tutorial: Indices (Explained with Examples)

This Mathematics Tutorial will focus on INDICES. We will explain it with examples and give you exercise. At the end of the tutorial, you’ll be able to download it for FREE. Please share this page with your friends who may need it.

In mathematics, indices, also known as exponents or powers, are a way of expressing the repeated multiplication of a number by itself.

We will break this down for you.

Let’s say you have a number like 2, and you want to multiply it by itself a certain number of times. Instead of writing out the multiplication several times, you can use indices to represent this more concisely.

Here’s how it works:

Base Number: The number you’re multiplying by itself is called the base number. In our example, the base number is 2.
Exponent: The small number written above and to the right of the base number is called the exponent. It tells you how many times to multiply the base number by itself.

Let me give you an example:

Here, 2 is the base number, and 3 is the exponent. It means you multiply 2 by itself three times:

2³ = 2×2×2 = 8

So, 2³ is equal to 8.

Let’s try another one:

Here, 5 is the base number, and 2 is the exponent. It means you multiply 5 by itself two times:

5²= 5×5 =25

So, 5² is equal to 25.

Now let’s look at an example with a negative base:

(−3)²

Here, -3 is the base number, and 2 is the exponent. It means you multiply -3 by itself two times:

(−3)² = (−3) × (−3) = 9

Notice that when we multiply two negative numbers, we get a positive result.
Also note that: A negative base raised to an odd index is negative. While a negative base raised to an even index is positive

Now, let’s look at a case with a negative exponent:

2⁻³

Here, 2 is the base number, and -3 is the exponent. It means you take the reciprocal of 2 raised to the positive 3:

2⁻³= ½³ = ⅛

In this case, a negative exponent indicates that we take the reciprocal¹ of the base number raised to the positive value of the exponent.

Page Contents

Laws of Indices
Exercises:
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Laws of Indices

The laws of indices, also known as the laws of exponents, are a set of rules that help simplify expressions involving powers. Here are the main laws of indices with examples:

Product Law: a^m × aⁿ = a^m+n
This law states that when you multiply two terms with the same base, you add the exponents.
Example: 2³×2⁴ = 2³⁺⁴ = 2⁷ =128
Quotient Law: a^m / aⁿ =a^m−n
This law states that when you divide two terms with the same base, you subtract the exponents.
Example: 3² / 3⁵ = 3⁵⁻² = 3³ =27
Power Law:(a^m)ⁿ =a^m−n
This law states that when you raise a power to another power, you multiply the exponents.
Example: (4²)³ =4^2×3 =4⁶ =4096
Negative Exponent Law: a^−m = 1 / a^m
This law states that a negative exponent is equivalent to the reciprocal of the term with a positive exponent.
Example: 2⁻³=1 / 2³ = 1 / 8
Zero Exponent Law: a⁰ = 1
Any nonzero number raised to the power of zero is equal to 1.
Example: 5⁰ = 1

These laws are fundamental in simplifying expressions involving powers and provide a systematic way to manipulate and solve problems involving exponents.

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Exercises:

(1) List the first six powers of (a) 2 (b) 3 (c) 4

(2) Copy and complete the values of these common powers
(a) 5¹
(b) 5⁴
(c) 6³
(d) 5²
(e) 6¹
(f) 7⁴

(3) Simplify
(a) (-1)⁵
(b) (-1)⁶
(c) (-2)⁵

(4) Simplify:
(a) 4^x = 8
(b) 9^x-2 = ⅓

SOLUTIONS TO EXERCISES

Make sure you try solving the exercises yourself. It’ll really help you understand the topic.

(1) List the first six powers of (a) 2 (b) 3 (c) 4

(a) First six powers of 2:

2¹=2
2²=4
2³=8
2⁴=16
2⁵=32
2⁶=64

(b) First six powers of 3:

3¹=3
3²=9
3³=27
3⁴=81
3⁵=243
3⁶=729

(c) First six powers of 4:

4¹=4
4²=16
4³=64
4⁴=256
4⁵=1024
4⁶=4096

(2) Copy and complete the values of these common powers
(a) 5¹
(b) 5⁴
(c) 6³
(d) 5²
(e) 6¹
(f) 7⁴

(a) 5¹: 5 x 1 = 5

(b) 5⁴: 5×5×5×5=625

(c) 6³: 6×6×6=216

(d) 5²: 5×5=25

(e) 6¹: 6×1=6

(f) 7⁴: 7×7×7×7=2401

These values represent the results of raising the given bases to the specified powers using indices.

(3) Simplify (a) (-1)⁵ (b) (-1)⁶ (c) (-2)⁵

(a) (−1)⁵ means multiplying -1 by itself 5 times. ie (−1)⁵ = −1 × −1 × −1 × −1 ×−1 = −1

(b) (−1)⁶ means multiplying -1 by itself 6 times. (−1)⁶ = −1× −1× −1× −1× −1× −1 = 1

(c) (−2) means multiplying -2 by itself 5 times. (−2)⁵ = −2× −2× −2× −2× −2 = −32

So, after simplifying:
(a) (−1)⁵ = −1,
(b) (−1)⁶ = 1,
(c) (−2)⁵ = −32.

(4) Simplify:
(a) 4^x = 8
(b) 9^x-2 = ⅓

(a) 4^x = 8
2^2x = 2³
Equate the powers, therefore
2x = 3,
x = 3/2 or 1½

(b) 9^x-2 = ⅓
(3²)^x-2 = 3^-1
Equate the powers,
2(x-2) = -1,
2x-4 = -1,
2x = -1+4
2x = 3,
x = 3/2

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Footnotes:

The reciprocal of a number is simply 1/number.
So, if you have a number like 2, the reciprocal of 2 is ½.
Similarly, the reciprocal of −3 is 1/−3.
In the case of 2⁻³, it means 1/2³, which is ⅛.
In summary, reciprocals express the multiplicative inverse of a number, meaning a number that, when multiplied by the original number, gives a product of 1. ↩︎

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