Powers of Real Numbers(Exponents)

Exponents represent repeated multiplication. For example,

Introduction

More generally, for any non-zero real number a and for any whole number n,

Introduction

In the exponential expression an, a is called the base and n is called the exponent.

a2 is read as ‘a squared’. a3 is read as ‘a cubed’. a4 is read as ‘a to the fourth power’. ... an is read as ‘a to the nth power’.

Caution!

Caution!

Properties of Exponents

Example:

Properties of Exponents

Example:

Properties of Exponents

Homework.

Example:

Some Definitions of Exponents

Properties of Exponents

Homework.

Example:

Properties of Exponents

Homework.

Example:

Properties of Exponents

Homework.

Example:

Properties of Exponents

1. All powers of a positive real number a are positive, i.e. for a ∈ R, a > 0, and n ∈ Z,

an > 0. 2. The even powers of a negative real number a are positive, i.e. for a ∈ R, a ≠ 0 and n ∈ Z,

(–a)n = an (if n is an even number).3. The odd powers of a negative real number a are negative, i.e. for a ∈ R, a ≠ 0 and n ∈ Z,

(–a)n = –an (if n is an odd number)

Example:

Example:

Properties of Exponents

The terms of an expression which have the same base and the same exponent are called like terms. We can add or subtract like terms.

(x ⋅ an) + (y ⋅ an) + (z ⋅ an) = (x + y + z) ⋅ an (a ≠ 0)

Example:

Let a ∈ R – {–1, 0, 1} (a is a real number other than –1, 0 and 1).

If am = an then m = n.

Exponential Equations

2x = 16

3x+1 = 81

22x + 1 = 8x – 1

Examples:

Exercises: