Indices ‖ Meaning, Laws And Worked Examples

This article will give you all the information you need to know about indices in mathematics, including their meaning, the Laws of Indices, and several worked examples. If you have difficulties solving arithmetic problems involving indices, this article will address that. Ensure to read to the end.

What Are Indices?         

An index of a number x is the number of times x is multiplied by itself. You can also refer to indices as the power of a number. Let us break it down further:

Consider that 3² is referred to as “three-squared” or three raised to the power of two. Simply put, it means multiplying three by itself in two places:

3² = 3 X 3

4⁴ = 4 X 4 X 4 X 4

You should note that the term “index, plural; indices” can be used interchangeably with “power”. Therefore, the index of a number is also the power of the number. Another important thing you should note in indices is the base number.

Consider 3², 3 X 3 = 9

3 is the base number, whereas the superscript 2 is the index. In the calculation above, the index form of 9 is 3². You can apply indices to integers, fractions, and even mixed numbers.

Examples Of Indices

Write the numbers below in their index forms:

1. 1/8
2. 216
3. 3125
4. 64
5. 144

Solutions

To solve these, you must find a number that can multiply itself (either once or severally) to give the given number.

1. 1/8 = ½ X ½ X ½ = (½)³
2. 216 = 6 X 6 X 6 = 6³
3. 3125 = 5 X 5 X 5 X 5 X 5 = 5⁵
4. 64 = 8 X 8 = 8²
5. 144 = 12 X 12 = 12²

Laws Of Indices

To solve basic arithmetic operations involving indices, you must follow a set of rules, otherwise known as laws. These laws are mathematical rules that govern the operation of indices.

There are six (6) basic Laws of Indices. We will list each of these laws and discuss them one after another in this section. The laws are:

1. Multiplication law
2. Division law
3. Power law
4. Fractional indices
5. Zero Index Law
6. Negative Indices

Multiplication Law

We can define this law as follows:

N^a = N X N X N X … a number of places

N^b = N X N X N X … b number of places

Then N^a x N^b = {N X N X N X … a number of places} X {N X N X N X … b number of places}

= {N X N X N X … (a + b) number of places}

= N^{a + b}

Therefore, to define this multiplication law, whenever you encounter two given numbers with the same base multiplying, write down the base and add their powers together.

Examples

Simplify the following problems (i) 2⁴ x 2²  (ii) 3³ x 3² (iii) 5² X 5 ⁴

Solutions

• 2⁴ x 2² = 2^{4 + 2} = 2⁶
• 3³ x 3² = 3^{3 + 2} = 3⁵
• 5² X 5 ⁴ = 5^{2 + 4} = 5⁶

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Division Law

This law is very similar to the multiplication law you learned above. However, in this law, rather than adding the powers, you subtract the divisors.

By definition, we can say:

N^a ÷ N^b = N^{a – b}

If a is greater than b (a>b), then you will get positive indices

If a is less than b (a<b), you will get negative indices

However, if a = b, you will get zero law indices.

Worked Examples

Simplify the following (i) 6^4a ÷ 6^a (ii) 4³ ÷ 4 (iii) 5³ – 5²

Solutions

• 6^4a ÷ 6^a

Here, you have to apply the basic division law

6^4a ÷ 6^a = 6^{4a – a}  = 6^3a

• 4³ ÷ 4

To solve this, you must recall: X¹ = X (any number raised to the power of one is that same number)

Therefore 4 = 4¹

4³ ÷ 4 = 4^{3 – 1}

Finally given as = 4²

• 5³ – 5² = 5^{3 – 2}

= 5¹

= 5

Power Law

Indices involving power law require that you multiply the powers. We can define it as follows

(N^a)^b = N^ab

By this definition above, you are required to multiply the powers of the base number. It is similar to when you want to open brackets in standard arithmetic calculations, but you only multiply the powers this time.

Worked Examples

Simplify the following (i) (3²)³  (ii) (5²)⁴ (iii) (2⁴)²

Solutions

• (3²)³

= 3^{2 x 3}

= 3⁶

• (5²)⁴

= 5^{2 x 4}

= 5⁸

• (2⁴)²

= 2^{4 x 2}

= 2⁸

Fractional Indices (N^1/a)

This is a type of index whose power is a fraction. We can define this type of indices as follows:

Recall;                   2^1/2  is given as √2 (the square root of 2)

2^1/3 is given as ³√2 (the cube root of 2)

Then,                    N^1/a  is given as ^a√N (the a^th root of N)

N^a/b is given as ^b√N^a (the b^th root of N^a)

Where N and a are any numbers, and N is a positive number. You have to note that fractional indices involve roots.

Worked Examples

Simplify the following: (i) 27^1/3 (ii) 144^1/2 (iii) ³√16²

Solution

• 27^1/3 = ³√27

= ³√3 X 3 X 3

Then you have, = ³√3³

= 3

• 144^1/2

⁼ √3² X 4²

The squares will cancel out with the square roots

= 3 X 4

= 12

• ³√16²

= ³√(2 X 2 X 2) X (2 X 2 X 2)

Then it proceeds to = ³√2³ X 2³

The cube will cancel out with the cube root

= 2 X 2

= 4

Zero Index Law

This is the simplest law of indices we have. It states that any number raised to the power of 0 = 1

By definition, we have:

N⁰ = 1

Worked Example

Simplify the following (i) 15⁰ (ii) 10⁰  (iii) 100⁰

Solution

• 15⁰ = 1
• 10⁰ = 1
• 100⁰ = 1

Negative Indices (N^-a)

By definition, this law is stated as:

N^-a = 1/N^a

Worked Examples

Simply the following (i) 2^-2 (ii) 5^-3

Solution

• 2^-2 = 1/2²

= 1/2 X 2

= ¼

• 5^-3

= 1/5³

Then it proceeds to 1/5 X 5 X 5

= 1/125

This article has provided a detailed explanation of indices, including the definition and laws of indices with worked examples. Please use our comment box below for further questions concerning indices. We will respond to you as soon as possible.

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