Questions: Use like bases to solve the exponential equation. n = (1/32)^(3n) * 8 = 2^6 Recall that exponential functions are one-to-one. In the given equation, are the bases of the exponents the same? If no the same base? When the bases are the same, use the one-to-one property to equate the exponents.

Solution Steps

To solve the given exponential equation, we need to express all terms with the same base. Here, we can use base 2 for all terms. Once the bases are the same, we can equate the exponents and solve for \( n \).

We start with the equation: \[ n = \left(\frac{1}{32}\right)^{3n} \cdot 8 = 2^6 \] We express all terms with base 2: \[ \left(2^{-5}\right)^{3n} \cdot 2^3 = 2^6 \]

This simplifies to: \[ 2^{-15n} \cdot 2^3 = 2^6 \] Combining the exponents gives: \[ 2^{-15n + 3} = 2^6 \]

Since the bases are the same, we can equate the exponents: \[ -15n + 3 = 6 \]

Rearranging the equation: \[ -15n = 6 - 3 \] \[ -15n = 3 \] Dividing both sides by -15: \[ n = -\frac{1}{5} \]

Final Answer