To rewrite \( 3\sqrt{5}^2 \) with a rational exponent, we first recognize that the square root can be expressed using a rational exponent. The square root of a number \( x \) can be written as \( x^{1/2} \).
Thus, we can rewrite \( 3\sqrt{5}^2 \) as:
\[ 3(\sqrt{5})^2 = 3(5^{1/2})^2 \]
Now, we apply the power of a power rule, which states \( (a^m)^n = a^{mn} \):
\[ 3(5^{1/2})^2 = 3 \cdot 5^{(1/2) \cdot 2} = 3 \cdot 5^1 \]
So, \( 3 \cdot 5^1 \) can also be written as:
\[ 3 \cdot 5 = 15 \]
Alternatively, if you want to keep the expression in terms of \( 5 \) in a rational exponent format, the final expression with rational exponents would be:
\[ 3 \cdot 5^{1} \]
Thus, the exact final answer, as the original form was a result, is \( 15 \) or in the form with the rational exponent \( 3 \cdot 5^{1} \).
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