## Definitions and Rules for Exponents

Definitions and Rules for Exponents

For all integers m and n and all real numbers a and b, the following rules apply.

Product Rule am.an = a m+n

Quotient Rule = am-n(a ≠ 0)

Zero Exponent a0 = 1 (a ≠ 0)

Negative Exponent a –n = (a ≠ 0)

Power Rules (am)n = amn

(ab)m = ambm

Special Rules

## Power Rules for Exponents

Power Rules for Exponents

If a and b are real numbers and m and n are integers, then

(a)  (am)n = amn .

(b)  (ab)m = ambm

(c)  (a/b)m = am/bm (b ≠0).

In words,

(a) to raise a power to a power, multiply exponents;

(b) to raise a product to a power, raise each factor to that power; and

(c) to raise a quotient to a power, raise the numerator and the denominator

to that power.

## Quotient Rule for Exponents

Quotient Rule for Exponents

If a is any nonzero real number and m and n are integers, then

am/an = am-n.

In words, when dividing powers of like bases, keep the same base and

subtract the exponent of the denominator from the exponent of the

denominator from the exponent of the numerator.

## Define 0 and negative exponents

Define 0 and negative exponents

So far we have discussed only positive exponents. Now we define 0 as an exponent. Suppose

we multiply 42by 40. By the product rule, extended to whole numbers,

42 . 40 = 42+0 = 42.

For the product rule to hold true, 40must equal 1, and so we define a0this

way for any nonzero real number a.

Zero Exponent

If a is any nonzero real number, then

a0= 1.

The expression 00is undefined.*

Negative Exponent

For any natural number n and any nonzero real number a,

a-n = 1/an.

## Product Rule for Exponents

Product Rule for Exponents

If m and n are natural numbers and a is any real number, then

am. an = am+n.

In words, when multiplying powers of like bases, keep the same base

## Laws of Exponents

Laws of Exponents

The same laws hold for rational-number exponents as for integer exponents.

We list them for review.

For any real number a and any rational exponents m and n:

a.  am.an = am+n In multiplying, we can add exponents if the

bases are the same.

b. = am.n In dividing, we can subtract exponents if the

bases are the same.

c. (am)n = am.n  To raise a power to a power, we can multiply

the exponents.

d. (ab)m = ambm  To raise a product to a power, we can raise

each factor to the power.

e. To raise a quotient to a power, we can raise

both the numerator and the denominator to

the power.

## Negative Rational Exponents

Negative Rational Exponents

Negative rational exponents have a meaning similar to that of negative integer

exponents.

a-m/n

For any rational number m/n and any positive real number a,

a-m/n means

that is, am/n and a-m/n are reciprocals.

## Rational Exponents

Rational Exponents

Expressions like  a1/2 , 5-1/4 ,and (2y)4/5 have not yet been defined. We will define

such expressions so that the general properties of exponents hold.

Consider a1/2.a1/2. If we want to multiply by adding exponents, it must

follow that a1/2 . a1/2 = a1/2 +1/2, or a1 Thus we should define a1/2 to be a

square root of a. Similarly, a1/3.a1/3 a1/3 = a1/3+1/3+1/3,or  a1,  so a1/3

should be defined to mean .

a1/n

For any nonnegative real number a and any natural number index n

(n ≠1)

a1/n  means   (the nonnegative nth root of a).

am/n

For any natural numbers m and n (n ≠1)and any nonnegative real

number a,

am/n means   , or