To find the exponent that makes the two expressions equivalent, we can set the two expressions equal to each other and then solve for the unknown exponent.
You provided the two expressions:
- \((16^{\pi})^6\)
- \(16^{x} \cdot 18^{y}\)
First, let's simplify the first expression:
\[ (16^{\pi})^6 = 16^{6\pi} \]
Now we can set the two expressions equal to each other:
\[ 16^{6\pi} = 16^{x} \cdot 18^{y} \]
Now, if we assume you want to compare the bases of 16 only, we can just work with the base \(16\) and try to express \(18\) in terms of \(16\). However, that might not be straightforward since 18 is not a power of 16.
Instead, we can let \(y\) be zero which simplifies our problem to just finding \(x\):
\[ 16^{6\pi} = 16^{x} \]
From the equation above, since the bases are the same, we can equate the exponents:
\[ 6\pi = x \]
Thus, the value of \(x\) that makes the two expressions equivalent is:
\[ \boxed{6\pi} \]
If you also want to consider \(18^y\), then you would have to plug in specific values for \(y\) to solve the equation properly as \(16^{6\pi} = 16^x \cdot 18^y\). For the simplest case as set above, \(y\) would be zero yielding \(6\pi\).
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