41.5 Powers with Integer Exponents
Powers with Integer Exponents explore how numbers grow or shrink through repeated multiplication, forming the foundation for more complex algebraic expressions.
Powers with Integer Exponents describes how an expression already raised to a power is itself raised to another integer power, covering the multiplication of exponents for a power-of-a-power, the distribution of an outer exponent across a product or quotient, and the special handling required when that outer exponent is zero or negative.
Integer Power-of-a-Power Rule
Rule Statement
When a power is itself raised to another power, the two exponents are multiplied together while the base remains unchanged:
Example
Integer Power of a Product
Rule Statement
When a product of two factors is raised to a power, that power is distributed to each factor individually:
Example
Integer Power of a Quotient
Rule Statement
When a quotient of two factors is raised to a power, that power is distributed to both the numerator and the denominator:
Example
Zero Outer Exponent Case
Description
When the outer exponent applied to a power, product, or quotient is exactly zero, the entire expression simplifies directly to one, provided the base or bases involved are nonzero, following the same zero-exponent rule established for a single base.
Example
Negative Outer Exponent Reciprocal Conversion
Description
When the outer exponent applied to a power, product, or quotient is negative, the entire expression is converted into a reciprocal using the positive version of that exponent, following the same negative-exponent rule established for a single base.
Example
Grouping Requirement for Powered Products
Requirement
The power-of-a-product rule applies only when the entire product is enclosed together and raised to the exponent as a single unit; an exponent attached to only one factor of an unenclosed expression does not distribute to the other factor.
Contrast
, where the entire product is grouped, differs from , where the exponent applies only to , and these two expressions are not equal.Grouping Requirement for Powered Quotients
Requirement
The power-of-a-quotient rule applies only when the entire quotient is enclosed together and raised to the exponent as a single unit; an exponent attached to only the numerator or only the denominator does not distribute to the other part.