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Power of a power rule examples: ( a 2 ) 3 = a 2 × 3 = a 6. ( 2 4 ) 2 = 2 4 × 2 = 2 8. [ ( p + q ) 5 ] 7 = ( p + q ) 5 × 7 = ( p + q ) 35. [ ( ? 1 ) 4 ] 2 = ( ? 1 ) 4 × 2 = ( ? 1 ) 8 = 1.
This leads to another rule for exponents?the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, (23)5=215 ( 2 3 ) 5 = 2 15 .
Laws of Exponents. When multiplying like bases, keep the base the same and add the exponents. When raising a base with a power to another power, keep the base the same and multiply the exponents. When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent.
If n is a positive integer and x is any real number, then xn corresponds to repeated multiplication xn=x×x×?×x?n times. We can call this ?x raised to the power of n,? ?x to the power of n,? or simply ?x to the n.? Here, x is the base and n is the exponent or the power.
Answer: 2 to the power 5 can be expressed as 25 = 2 × 2 × 2 × 2 × 2 = 32.