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Exponents and Roots Help

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

Multiplication and Division with Negative Numbers Examples

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

Examples

Multiplication and Division with Negative Numbers 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.

Multiplication and Division with Negative Numbers Examples

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

Multiplication and Division with Negative Numbers 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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