Technical Mathematics, Sixth Edition

2.3. Laws of Exponents

2.3.1. Definitions

We have done some work with powers of numbers in Chapter 1. Now we will expand those ideas to include powers of algebraic expressions. Here we will deal only with expressions that have integers (positive or negative, and zero) as exponents. We will need the laws of exponents to multiply and divide algebraic expressions in the following sections.

Recall from Chapter 1 that a positive exponent shows how many times the base is to be multiplied by itself.

Example 32: In the expression 25, the base is 2 and the exponent is 5.

In general,

NOTE

NOTE

An exponent applies only to the symbol directly in front of it.

Thus,

5y³ = 5(y³)

but

5y³ ≠ 5³y³

2.3.2. Multiplying Powers

▪ Exploration:

Try this. (a) Use Eq. 21 to expand x³ into its factors, that is, (x)(x)(x). (b) Similarly expand x⁴ into its factors. (c) Form the product of x³ and x⁴, with each in its factored form. (d) Simplify your expression and use Eq. 21 again to write your result as x raised to a power.

What did you find? Can you express your findings as a rule?

You should have found that the exponent in the product is the sum of the two original exponents.

x³ · x⁴ = x³⁺⁴ = x⁷

This gives our first law of exponents.

NOTE

Example 33: Here we show the use of Eq. 21. x2(x5) = x2+5 ...