Properties of exponents

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exponent

shows repeated multiplication of a

base.

For example, in the expression

is the base and 4 is the exponent. You can expand the expression by using the base,

as a factor four times.

You can refer to an exponent expression such as

as a

power.

Properties of exponents can help to simplify expressions with exponents. The table shows some properties of exponents and examples.

Property name	Property	Example
Product of powers	x n x m = x n + m	3 2 3 5 = 3 7
Quotient of powers	x n x m = x n – m	4 9 4 7 = 4 2
Power of a power	( x n ) m = x n m	( 5 2 ) 6 = 5 12
Power of a product	( x y ) n = x n y n	( 2 3 ) 4 = 2 4 3 4
Power of a quotient	i x y i n = x n y n	i 4 5 i 7 = 4 7 5 7
Zero exponent	x 0 = 1	3 0 = 1
Negative exponent	x – n = 1 x n	4 – 6 = 1 4 6

Product of powers

You can multiply powers with the same base using the product of powers

property.

To multiply powers with the same base, keep the base and add the exponents.

Let's try it! Simplify

So,

To see why the product of powers property works, expand each power.

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Try some practice problems!

Multiply powers: integer bases

Quotient of powers

You can divide powers with the same base using the quotient of powers

property.

To divide powers with the same base, keep the base and subtract the exponents.

–

Let's try it! Simplify

–

So,

To see why the quotient of powers property works, expand each power.

Try some practice problems!

Divide powers: integer bases

Power of a power

You can find the power of a power using the power of a power

property.

To find the power of a power, multiply the exponents.

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Let's try it! Simplify

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So,

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To see why the power of a power property works, expand

and then use that expansion as a factor four times.

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This property is also known as the power rule.

Try some practice problems!

Power of a power: integer bases

Power of a product

You can find the power of a product using the power of a product

property.

To find the power of a product, multiply the powers of the individual factors.

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Let's try it! Simplify

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To see why the power of a product property works, expand

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Power of a quotient

You can find the power of a quotient using the power of a quotient

property.

To find the power of a quotient, divide the powers of the numerator and denominator.

Let's try it! Simplify

To see why the power of a quotient property works, expand

Zero power

You can evaluate nonzero bases raised to the zero power using the zero exponent

property.

Any nonzero base to the zero power is 1.

Let's try it! Evaluate

Let's see why the zero exponent property works. Start by simplifying

using the quotient of powers property.

(

–

)

So,

Now, evaluate

Remember that any number divided by itself is equal to 1.

Since

and

then

Negative exponents

You can simplify negative exponents using the negative exponent

property.

When you see a base with a negative exponent, you can turn it into a fraction with a numerator of 1 and a positive exponent in the denominator.

–

Let's try it! Simplify

–

Let's see why the negative exponent property works. Start by simplifying

using the quotient of powers property.

–

Now, simplify

by expanding each power.

Since

–

and

then

–

Try some practice problems!

Evaluate powers with negative exponents

Skills

Evaluate powers with negative exponents Multiply powers: integer bases Divide powers: integer bases Power of a power: integer bases Evaluate expressions using properties of exponents Identify equivalent expressions involving exponents II

Videos(6)

Evaluate powers with negative exponents

Multiply powers: integer bases

Divide powers: integer bases

Power of a power: integer bases

Evaluate expressions using properties of exponents

Identify equivalent expressions involving exponents II

Lessons

Exponents

Product of powers

Try some practice problems!

Quotient of powers

Try some practice problems!

Power of a power

Try some practice problems!

Power of a product

Power of a quotient

Zero power

Negative exponents

Try some practice problems!

Related

Skills

Videos(6)

Lessons