**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.

**Examples**

**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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