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 2^{5}, 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,
 5y^{3} = 5(y^{3})
but
 5y^{3} ≠ 5^{3}y^{3}
2.3.2. Multiplying Powers
▪ Exploration:
Try this. (a) Use Eq. 21 to expand x^{3} into its factors, that is, (x)(x)(x). (b) Similarly expand x^{4} into its factors. (c) Form the product of x^{3} and x^{4}, 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.
 x^{3} · x^{4} = x^{3+4} = x^{7}
This gives our first law of exponents.
NOTE
Example 33:Here we show the use of Eq. 21.

Get Technical Mathematics, Sixth Edition now with the O’Reilly learning platform.
O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.