Solving the Equation: 1/64 × 4^x × 2^x = 64
In this article, we will solve the equation 1/64 × 4^x × 2^x = 64. This equation involves exponential functions and requires some algebraic manipulation to find the value of x.
Step 1: Simplify the Left-Hand Side
Let's start by simplifying the left-hand side of the equation:
1/64 × 4^x × 2^x = ?
We can rewrite 4 as 2^2, so we get:
1/64 × (2^2)^x × 2^x = ?
Using the property of exponentiation, we can rewrite the equation as:
1/64 × 2^(2x) × 2^x = ?
Now, we can combine the two exponential terms:
1/64 × 2^(3x) = ?
Step 2: Equate the Expression to 64
Now that we have simplified the left-hand side, we can equate it to 64:
1/64 × 2^(3x) = 64
Step 3: Solve for x
To solve for x, we can start by multiplying both sides of the equation by 64:
2^(3x) = 64 × 64
2^(3x) = 4096
Now, we can take the logarithm of both sides of the equation (base 2) to get:
3x = log2(4096)
3x = 12
x = 12/3
x = 4
Conclusion
Therefore, the value of x is 4. We have successfully solved the equation 1/64 × 4^x × 2^x = 64.
