We start by rewriting both sides of the equation \( 16^x = \frac{1}{64} \) as powers of 2:

\[
16 = 2^4 \quad \text{and} \quad 64 = 2^6
\]

So, the equation becomes:

\[
(2^4)^x = \frac{1}{2^6}
\]

Using the power of a power rule, \( (a^m)^n = a^{m \cdot n} \), we simplify the left-hand side:

\[
2^{4x} = \frac{1}{2^6}
\]

The right-hand side, \( \frac{1}{2^6} \), can be written as \( 2^{-6} \), so the equation becomes:

\[
2^{4x} = 2^{-6}
\]

Since the bases are the same, we can set the exponents equal to each other:

\[
4x = -6
\]

Now, solve for \( x \):

\[
x = \frac{-6}{4} = -\frac{3}{2}
\]

Thus, the solution is:

\[
x = -\frac{3}{2}
\]

Would you like more details on any part of the solution?

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Here are 5 related questions:

1. How do you solve \( 2^x = \frac{1}{32} \)?
2. What is the solution to \( 9^x = \frac{1}{81} \)?
3. How do you express \( 8^x = \frac{1}{512} \) using powers of 2?
4. What is the value of \( x \) if \( 4^x = 1/16 \)?
5. How would you solve \( 27^x = 1/243 \) using powers of 3?

**Tip:** When solving exponential equations, it helps to express all terms with the same base to simplify the comparison of exponents.

Learn how to solve the exponential equation 16^x = 1/64 by expressing both sides as powers of 2. This detailed step-by-step solution involves exponent rules and simplifies the equation by comparing the exponents. Suitable for students in Grades 9-11.

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