## 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.

## 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.

## TO FACTOR WHEN CONSTANT TERM POSITIVE OR NEGATIVE

TO FACTOR  WHEN CONSTANT TERM POSITIVE OR NEGATIVE

TO FACTOR  x2 + bx + c WHEN  IS  POSITIVE

When the constant term of a trinomial is positive, look for two numbers with the same sign. The sign is that of the middle term:

x2 – 7x + 10 = (x – 2) (x – 5);

x2 + 7x + 10 = (x + 2) (x + 5).

TO FACTOR  x2 + bx + c WHEN  IS  NEGATIVE

When the constant term of a trinomial is negative, look for two numbers whose product is negative. One must be positive and the other negative:

x– 4x – 21 = (x + 3) (x – 7);

x+ 4x – 21 = (x – 3) (x + 7).

Consider pairs of numbers for which the number with the larger absolute value has the same sign as b, the coefficient of the middle term.