Understanding the Problem
To solve the equation 4^2x = 1/32, we start by rewriting the numbers in terms of powers of 2.
Step 1: Rewrite 4 and 32
- The base 4 can be expressed as 2^2. Therefore, 4^2x can be rewritten as (2^2)^(2x) = 2^(4x).
- The number 32 is a power of 2, specifically, 32 = 2^5. Therefore, 1/32 can be expressed as 2^(-5).
Step 2: Equate the Exponents
Now we can set the equations equal to each other:
- 2^(4x) = 2^(-5)
Since the bases are the same, we can equate the exponents:
- 4x = -5
Step 3: Solve for x
To find x, divide both sides by 4:
- x = -5/4
Final Answer
Thus, the value of x is:
- x = -1.25
This means that if 4^2x = 1/32, then the solution to the equation is x = -1.25.
Conclusion
Understanding how to manipulate exponents and rewrite numbers in terms of common bases is key to solving such equations effectively.