Find the value of m for which 9^m ÷ 3^(-2) = 9^4.

To solve for the value of m in the equation 9 m ÷ 3 − 2 = 9 4 , let's break down the problem step-by-step using the properties of exponents.

Identify the expressions using powers of 3:


We know that 9 can be expressed as 3 2 . Therefore, 9 m = ( 3 2 ) m = 3 2 m .


Rewrite the division operation as multiplication:


When dividing by a power, we can multiply by its reciprocal: 3 2 m ÷ 3 − 2 = 3 2 m × 3 2


Apply the product of powers property:


The product of powers property states that a m × a n = a m + n . Using this property, we can combine the exponents: 3 2 m × 3 2 = 3 2 m + 2


Set the expression equal to the right-hand side of the equation:


We now have: 3 2 m + 2 = 9 4

But since 9 4 = ( 3 2 ) 4 = 3 8 , we can equate the exponents: 2 m + 2 = 8



Solve for m :


Subtract 2 from both sides: 2 m = 6

Divide both sides by 2: m = 3


Thus, the value of m that satisfies the equation 9 m ÷ 3 − 2 = 9 4 is 3 .

Answered by BenjaminOwenLewis | 2025-07-06