Question

81^5 = 3^x
x =
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Explanation:

Step1: Express 81 as a power of 3

We know that \( 81 = 3^4 \), so we can rewrite the left - hand side of the equation \( 81^{5}=3^{x} \) as \( (3^{4})^{5} \).

Step2: Use the power - of - a - power rule

The power - of - a - power rule states that \( (a^{m})^{n}=a^{m\times n} \). For \( (3^{4})^{5} \), where \( a = 3 \), \( m = 4 \) and \( n = 5 \), we have \( (3^{4})^{5}=3^{4\times5}=3^{20} \).
Since \( 81^{5}=(3^{4})^{5}=3^{20} \) and \( 81^{5}=3^{x} \), then \( 3^{20}=3^{x} \).
If two exponential expressions with the same base are equal, then their exponents are equal. So, \( x = 20 \).

Answer:

\( 20 \)