Question

What exponent would make the two
expressions equivalent: (16x) to the power of 6 = (16 to the power of 18)

Answer 1

To solve for the exponent that makes the two expressions equivalent, we need to rewrite both expressions in terms of powers of 16.

The left side, \((16x)^6\), can be rewritten using the property of exponents:

\[
(16x)^6 = 16^6 \cdot x^6
\]

Now, the right side is already given as \(16^{18}\).

We want to determine when these two expressions are equivalent:

\[
16^6 \cdot x^6 = 16^{18}
\]

To solve for \(x\), we can first isolate \(x^6\) by dividing both sides by \(16^6\):

\[
x^6 = \frac{16^{18}}{16^6}
\]

Using the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\), we have:

\[
x^6 = 16^{18 - 6}
\]

Calculating \(18 - 6\) gives:

\[
x^6 = 16^{12}
\]

Now, to find \(x\), we take the sixth root of both sides:

\[
x = (16^{12})^{1/6} = 16^{12/6} = 16^2
\]

Therefore, we find that \(x = 16^2\) is the solution that makes the two expressions equivalent. To summarize, the exponent \(2\) makes the two expressions equivalent when \(x\) is substituted in. Thus, the two expressions are equivalent when:

\[
x = 256 \quad (\text{since } 16^2 = 256)
\]