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# Exponents and Roots Help

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By — McGraw-Hill Professional
Updated on Sep 26, 2011

## Exponents

The expression 4 x is shorthand for x + x + x + x , that is x added to itself four times. Likewise x 4 is shorthand for x·x·x·x—x multiplied by itself four times. In x 4 , x is called the base and 4 is the power or exponent . We say “x to the fourth power” or simply “x to the fourth.” There are many useful exponent properties. For the rest of the chapter, a is a nonzero number.

### Property 1 - Multiplying Like Bases

a n a m = a m+n

When multiplying two powers whose bases are the same, add the exponents.

#### Examples

2 3 · 2 4 = (2·2·2)(2·2·2·2) = 2 7    x 9 · x 3 = x 12

### Property 2 - Dividing Like Bases

When dividing two powers whose bases are the same, subtract the denominator’s power from the numerator’s power.

### Property 3 - Quantities Powered

( a n ) m = a nm

If you have a quantity raised to a power then raised to another power, multiply the exponents.

#### Examples

(5 3 ) 2 = (5·5·5) 2 =(5·5·5)(5·5·5)= 5 6    ( x 6 ) 7 = x (6)(7) = x 42

Be careful, Properties 1 and 3 are easily confused.

### Property 4 - Zero Power

a 0 = 1

Any nonzero number raised to the zero power is one. We will see that this is true by Property 2 and the fact that any nonzero number over itself is one.

Find practice problems and solutions at Exponents and Roots Practice Problems - Set 1.

### Exponent Properties in Algebra

These properties also work with algebraic expressions.

#### Examples

Be careful not to write (3 x – 4) 2 as (3 x ) 2 – 4 2 —we will see later that (3 x – 4) 2 is 9 x 2 – 24 x + 16.

Find practice problems and solutions at Exponents and Roots Practice Problems - Set 2.

More practice problems for this concept can be found at: Exponents and Roots Practice Test.

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