# Powers, Exponents, and Roots

## Negative Exponents

### Negative Exponents

Taking a number to a negative exponent does not necessarily yield a negative answer. Taking a base number to a negative exponent is equivalent to taking the base number to the positive opposite of the exponent (the exponent with the negative sign removed) and placing the result in the denominator of a fraction whose numerator is 1. For example, 5-4 = 1/54 = 1/625. 6-3 = 1/63 = 1/216, and (- 3)-2 = 1/(- 3)2 = 1/9.

If the base number is a fraction, then the negative exponent switches the numerator and the denominator. For example, (2/3)-4 = (3/2)4 = (34)/(24) = 81/16 and (- 5/6)-3 = (6/(- 5))3 = (63)/((- 5)3) = 216/(- 125) = - 216/125.

### Negative Exponents and the Base Ten System

Here is a list of negative powers of ten:

 10-1 = 1/101 = 1/10 = 0.1 10-2 = 1/102 = 1/100 = 0.01 10-3 = 1/103 = 1/1, 000 = 0.001 10-4 = 1/104 = 1/10, 000 = 0.0001 10-5 = 1/105 = 1/100, 000 = 0.00001

and so on...

Just as 102 represents a 1 in the hundreds place, 10-2 represents a 1 in the hundredths place. The single-digit number in the hundredths place is the number that is multiplied by 10-2.

Now we can write out any terminating decimal as a sum of single- digit numbers times powers of ten. The number 23.45 has a 2 in the tens place (2×101), a 3 in the ones place (3×100), a 4 in the tenths place (4×10-1) and a 5 in the hundredths place (5×10-2). Thus, 23.45 = 2×101 +3×100 +4×10-1 +5×10-2.

Examples: Write out the following numbers as single-digit numbers times powers of ten:

523.81 = 5×102 +2×101 +3×100 +8×10-1 +1×10-2

3.072 = 3×100 +0×10-1 +7×10-2 +2×10-3

46.904 = 4×101 +6×100 +9×10-1 +0×10-2 +4×10-3