To find the exponent \( x \) that makes the two expressions equivalent, we start with the equation:
\[ (16^x)^6 = (6^{18}) \]
We can simplify the left side using the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[ 16^{6x} = 6^{18} \]
Next, we need to express the bases in a common form. We know that \( 16 = 2^4 \), so we can rewrite the left side:
\[ (2^4)^{6x} = 6^{18} \]
Now, apply the power of a power exponent rule again:
\[ 2^{24x} = 6^{18} \]
Next, we can express \( 6 \) in terms of its prime factors. We have \( 6 = 2 \cdot 3 \), so we can write:
\[ 6^{18} = (2 \cdot 3)^{18} = 2^{18} \cdot 3^{18} \]
Now, we compare the expressions on both sides:
\[ 2^{24x} = 2^{18} \cdot 3^{18} \]
For these two sides to be equal, the powers of \( 2 \) must be equal and the powers of \( 3 \) must also align accordingly. However, since the left side has no factor of \( 3 \), we can set up our equations from the terms involving \( 2 \). This means we only equate the exponents of \( 2 \):
\[ 24x = 18 \]
Now, solve for \( x \):
\[ x = \frac{18}{24} = \frac{3}{4} \]
Thus, the exponent \( x \) that makes the two expressions equivalent is
\[ \boxed{\frac{3}{4}} \]
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