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Express the given expression without logs, in simplest form. Assume all variables represent positive values.

(3^(-3log_(3)4x^(2)))
Answer:

Express the given expression without logs, in simplest form. Assume all variables represent positive values. $\newline$ $\left(3^{-3 \log _{3} 4 x^{2}}\right)$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

Q. Express the given expression without logs, in simplest form. Assume all variables represent positive values.

\newline

\left(3^{-3 \log _{3} 4 x^{2}}\right)

\newline

Answer:

Understand and Apply Power Rule: Understand the given expression and apply the power rule of logarithms. $\newline$ The expression is $3^{-3\log_{3}4x^{2}}$ . According to the power rule of logarithms, $a^{\log_{a}(b)} = b$ . We will use this rule to simplify the expression.
Apply Power Rule to Logarithmic Part: Apply the power rule to the logarithmic part of the expression. $\newline$ The power rule states that $a^{(n\log_a(b))} = b^n$ . Here, $a = 3$ , $n = -3$ , and $\log_a(b) = \log_{3}4x^{2}$ . So we can rewrite the expression as $(4x^{2})^{-3}$ .
Simplify Using Power of a Power Rule: Simplify the expression using the power of a power rule. $\newline$ The power of a power rule states that $(b^n)^m = b^{n*m}$ . Applying this rule, we get $4^{-3} \times (x^{2})^{-3}$ .
Calculate Powers to Simplify: Simplify the expression further by calculating the powers. $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$ and $(x^{2})^{-3} = \frac{1}{(x^{2*3})} = \frac{1}{x^6}$ . So the expression becomes $\frac{1}{64} * \frac{1}{x^6}.$
Combine Fractions for Final Expression: Combine the fractions to get the final simplified expression. $\newline$ Multiplying the fractions, we get $\frac{1}{64x^6}$ .

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