Review the common properties of exponents that allow us to rewrite powers in different ways. For example, x²⋅x³ can be written as x⁵.
PropertyExample
x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript2, start superscript, 3, end superscript, dot, 2, start superscript, 5, end superscript, equals, 2, start superscript, 8, end superscript
start fraction, x, start superscript, n, end superscript, divided by, x, start superscript, m, end superscript, end fraction, equals, x, start superscript, n, minus, m, end superscriptstart fraction, 3, start superscript, 8, end superscript, divided by, 3, start superscript, 2, end superscript, end fraction, equals, 3, start superscript, 6, end superscript
left parenthesis, x, start superscript, n, end superscript, right parenthesis, start superscript, m, end superscript, equals, x, start superscript, n, dot, m, end superscriptleft parenthesis, 5, start superscript, 4, end superscript, right parenthesis, start superscript, 3, end superscript, equals, 5, start superscript, 12, end superscript
left parenthesis, x, dot, y, right parenthesis, start superscript, n, end superscript, equals, x, start superscript, n, end superscript, dot, y, start superscript, n, end superscriptleft parenthesis, 3, dot, 5, right parenthesis, start superscript, 7, end superscript, equals, 3, start superscript, 7, end superscript, dot, 5, start superscript, 7, end superscript
left parenthesis, start fraction, x, divided by, y, end fraction, right parenthesis, start superscript, n, end superscript, equals, start fraction, x, start superscript, n, end superscript, divided by, y, start superscript, n, end superscript, end fractionleft parenthesis, start fraction, 2, divided by, 3, end fraction, right parenthesis, start superscript, 5, end superscript, equals, start fraction, 2, start superscript, 5, end superscript, divided by, 3, start superscript, 5, end superscript, end fraction
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Product of powers

This property states that when multiplying two powers with the same base, we add the exponents.
x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript

Example

5, start superscript, 2, end superscript, dot, 5, start superscript, 5, end superscript, equals, 5, start superscript, 2, plus, 5, end superscript, equals, 5, start superscript, 7, end superscript

Practice

Problem 1.1
Simplify.
Rewrite the expression in the form 8, start superscript, n, end superscript.
8, start superscript, 6, end superscript, dot, 8, start superscript, 4, end superscript, equals

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Quotient of powers

This property states that when dividing two powers with the same base, we subtract the exponents.
start fraction, x, start superscript, n, end superscript, divided by, x, start superscript, m, end superscript, end fraction, equals, x, start superscript, n, minus, m, end superscript

Example

start fraction, 3, start superscript, 8, end superscript, divided by, 3, start superscript, 2, end superscript, end fraction, equals, 3, start superscript, 8, minus, 2, end superscript, equals, 3, start superscript, 6, end superscript

Practice

Problem 2.1
Simplify.
Rewrite the expression in the form 7, start superscript, n, end superscript.
start fraction, 7, start superscript, 7, end superscript, divided by, 7, start superscript, 3, end superscript, end fraction, equals

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Power of a power property

This property states that to find a power of a power we multiply the exponents.
left parenthesis, x, start superscript, n, end superscript, right parenthesis, start superscript, m, end superscript, equals, x, start superscript, n, dot, m, end superscript

Example

left parenthesis, 8, start superscript, 2, end superscript, right parenthesis, start superscript, 3, end superscript, equals, 8, start superscript, 2, dot, 3, end superscript, equals, 8, start superscript, 6, end superscript

Practice

Problem 3.1
Simplify.
Rewrite the expression in the form 2, start superscript, n, end superscript.
left parenthesis, 2, start superscript, 4, end superscript, right parenthesis, start superscript, 2, end superscript, equals

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Power of a product

This property states that when taking the power of a product, we multiply the powers of the factors.
left parenthesis, x, dot, y, right parenthesis, start superscript, n, end superscript, equals, x, start superscript, n, end superscript, dot, y, start superscript, n, end superscript

Example

left parenthesis, 3, dot, 5, right parenthesis, start superscript, 6, end superscript, equals, 3, start superscript, 6, end superscript, dot, 5, start superscript, 6, end superscript

Practice

Problem 4.1
Select the equivalent expression.
left parenthesis, 4, dot, 7, right parenthesis, start superscript, 8, end superscript, equals, question mark
Choose 1 answer:
Choose 1 answer:

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Power of a quotient

This property states that when taking the power of a quotient, we divide the powers of the numerator and of the denominator.
left parenthesis, start fraction, x, divided by, y, end fraction, right parenthesis, start superscript, n, end superscript, equals, start fraction, x, start superscript, n, end superscript, divided by, y, start superscript, n, end superscript, end fraction

Example

left parenthesis, start fraction, 7, divided by, 2, end fraction, right parenthesis, start superscript, 8, end superscript, equals, start fraction, 7, start superscript, 8, end superscript, divided by, 2, start superscript, 8, end superscript, end fraction

Practice

Problem 5.1
Select the equivalent expression.
left parenthesis, start fraction, 6, divided by, 5, end fraction, right parenthesis, start superscript, 9, end superscript, equals, question mark
Choose 1 answer:
Choose 1 answer:

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