Indices

Index of a variable (or a constant) is a value that is raised to the power of the variable. The indices are also known as powers or exponents. It shows the number of times a given number has to be multiplied.

An index has a base (the number being multiplied) and a power or exponent (the number of times the number is multiplied).

An index is in the form Base^Power

Example, 2 x 2 x 2 x 2 x 2 could be written as 2⁵

The base is 2, the number being multiplied and the power is 5, the number of times the number (2) is multiplied.

Laws of indices

1. Multiplication Rule

This says that when indices are multiplying and the bases are the same, the result is the same as the base to the power of summation (addition) of the powers.

Base^Power x Base^Power = Base^{Summation (addition) of the powers}

a^m x aⁿ = a^m+n

NOTE: This law applies only when the bases are the same.

Examples

1. 2⁴ x 2² = 2⁴⁺² = 2⁶

2. 5¹ x 5² = 5¹⁺² = 5³

3. 3² x 3³ = 3²⁺³ = 3⁵

2. Dvision Rule

This says that when one index divides another index and their bases are the same, the result is the base to the power of subtraction of the power of the denominator from the power of the numerator.

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^m-n

NOTE: This law applies only when the bases are the same.

Examples

1. $\frac{2^4}{2^2}$ = 2^4-2 = 2²

2. 5¹ ÷ 5² = 5^1-2 = 5^-1

3. $\frac{3^5}{3^3}$ = 3^5-3 = 3²

3. Base raised to the power zero (0)

Every number or letter raised to the power zero (0) is 1

Base⁰ = 1

Examples

1⁰ = 1
2⁰ = 1
3⁰ = 1
4⁰ = 1
5⁰ = 1
6⁰ = 1
a⁰ = 1
b⁰ = 1
c⁰ = 1

4. Base with negative power

An index raised to a negative power is the same as 1 over the base to the positive power of that index

Base^-Power = $\frac{1}{{\mathrm{Base}}^{\mathrm{Power}}}$

Examples

1. 2^-1 = $\frac{1}{2^1}$

2. 2^-3 = $\frac{1}{2^3}$

5. Every number is raised to the power 1

Every number has a power of 1

a = a¹
b = b¹
2 = 2¹
3 = 3¹

6. Power outside a bracket

The power outside a bracket affects each term or expression in the bracket

Examples

(a x b)ⁿ = (a¹ x b¹)ⁿ = a^{1 x n} x b^{1x n} = aⁿ x bⁿ

(a^m x bⁿ)^c = a^{m x c} x b^{n x c}

($\frac{\mathrm{a}}{\mathrm{b}}$)^c = $\frac{{\mathrm{a}}^{\mathrm{c}}}{{\mathrm{b}}^{\mathrm{c}}}$

Examples

1. (2³ x 3⁴)² = 2^3x2 x 3^{4 x 2} = 2⁶ x 3⁸

2. ($\frac{3}{5}$)² = $\frac{3^2}{5^2}$

3. ($\frac{3^{\mathrm{½}}}{5}$)² = $\frac{3^{\mathrm{½\; x\; 2}}}{5^2}$ = $\frac{3^1}{5^2}$

7. Square of a number

The square of a number is the number multiplying itself

a² = a x a

Examples

2² = 2 x 2 = 4
3² = 3 x 3 = 9
4² = 4 x 4 = 16
5² = 5 x 5 = 25
6² = 6 x 6 = 36
7² = 7 x 7 = 49
8² = 8 x 8 = 64
9² = 9 x 9 = 81
10² = 10 x 10 = 100
11² = 11 x 11 = 121
12² = 12 x 12 = 144
13² = 13 x 13 = 169

8. The square root of a number

The result of a square root (√) is a number when multiplied by itself will give the number whose square root is being calculated.

Examples

√4 = 2 because 2 x 2 = 4
√9 = 3 because 3 x 3 = 9
√16 = 4 because 4 x 4 = 16
√25 = 5 because 5 x 5 = 25
√36 = 6 because 6 x 6 = 36
√49 = 7 because 7 x 7 = 49
√64 = 8 because 8 x 8 = 64
√81 = 9 because 9 x 9 = 81
√100 = 10 because 10 x 10 = 100
√121 = 11 because 11 x 11 = 121
√144 = 12 because 12 x 12 = 144
√169 = 13 because 13 x 13 = 169

9. Base to the power $\frac{1}{2}$

Any base raised to the power $\frac{1}{2}$ is the square root of that base

Examples

1. 4^½ = √4 = 2

2. 6^½ = √6

3. 9^½ = √9 = 3

4. 25^½ = √25 = 5

APPLICATION OF KNOWLEDGE

Question 1

If a = 2² x 2³ ÷ 2⁴

A. 2⁹ B. 2⁵ C. 2² D. 2

[B.E.C.E 1990 Section A Question 13]

Solution

Question 2

Simplify 2ab² x 3a²b.

A. 5a³b³ B. 5a²b² C. 6a³b³ D. 6a²b²

[B.E.C.E 1991 Section A Question 13]

Solution

Question 3

If a⁶ ÷ a⁴ = 64. Find a.

A. 8 B. 10 C. 16 D. 20

[B.E.C.E 1991 Section A Question 23]

Solution

Question 4

Find the value of √6$\frac{1}{4}$

A. 5.0 B. 4.9 C. 2.5 D. 2.4

[B.E.C.E 1992 Section A Question 4]

Solution

Question 5

Simplify 15⁹ ÷ 15⁷

A. B. C. D.

[B.E.C.E 1994 Section A Question 10]

Solution

Question 6

Simplify $\frac{6^2}{2^2\mathrm{x\; 3}}$

A. 1 B. 2 C. 3 D. 6

[B.E.C.E 1996 Section A Question 20]

Solution

Question 7

Simplify 35x⁵y ÷ 7xy²

A. 5x⁶y⁵ B. 5x⁴y^-1 C. 5x⁶y D. 5x⁴y⁵

[B.E.C.E 1997 Section A Question 21]

Solution

Question 8

What is the equivalent of 2² x 6² in terms of indices?

A. 2 x 3⁴ B. 2 x 3 C. 2⁴ x 3² D. 2⁴ x 3

[B.E.C.E 1998 Section A Question 18]

Solution

Question 9

Simplify 2 x 3² x 3⁴

A. 2 x 3⁵ B. 2 x 3⁶ C. 2 x 3⁸ D. 2 x 9²

[B.E.C.E 1999 Section A Question 13]

Solution

Question 10

Simplify $\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$

A. $\frac{2^2}{3^4}$ B. $\frac{3^4}{2^2}$ C. 3³ x 2⁴ D. $\frac{2^4}{3^4}$

[B.E.C.E 2004 Section A Question 9]

Solution

Question 11

Evaluate $\frac{2^{3}x{3}^{4}x{3}^3}{2^{2}x2x{3}^5}$

A. 6 B. 9 C. 12 D. 18

[B.E.C.E 2006 Section A Question 22]

Solution

Question 12

Simplify (2⁶ x 3⁴) ÷ (2⁴ x 3²).

A. 2² x 3² B. 2² x 3⁶ C. 2¹⁰ x 3² D. 2¹⁰ x 3⁶

[B.E.C.E 2007 Section A Question 35]

Solution

Question 13

Find the value of 4 + x⁰.

A. 3 B. 4 C. 5 D. 6

[B.E.C.E 2009 Section A Question 17]

Solution

Question 14

Evaluate $\frac{2^{7}x{3}^{4}x{5}^3}{2^{3}x{3}^{2}x{5}^2}$, leaving your answer in standard form.

[B.E.C.E 2010 Section B Question 3(a)]

Solution

Question 15

Simplify (8x²y³)($\frac{3}{8}$xy⁴)

[B.E.C.E 2019 Section B Question 2(c)]

Solution