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

(11^(log_(11)(sqrty)-log_(11)(z^(2))))
Answer:

Express the given expression without logs, in simplest form. Assume all variables represent positive values. $\newline$ $\left(11^{\log _{11}(\sqrt{y})-\log _{11}\left(z^{2}\right)}\right)$ $\newline$ 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(11^{\log _{11}(\sqrt{y})-\log _{11}\left(z^{2}\right)}\right)

\newline

Answer:

Apply Logarithmic Property: We are given the expression $11^{(\log_{11}(\sqrt{y}) - \log_{11}(z^{2}))}$ . To simplify this expression, we will use the properties of logarithms. $\newline$ The first property we will use is that $a^{(\log_{a}(b))} = b$ , where $a$ is the base of the logarithm and $b$ is the argument of the logarithm.
Combine Logarithmic Terms: We will also use the property that $\log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right)$ , which allows us to combine the two logarithmic terms into a single term.
Rewrite Using Combined Logarithm: Combining the logarithmic terms, we get: $\log_{11}(\sqrt{y}) - \log_{11}(z^{2}) = \log_{11}(\frac{\sqrt{y}}{z^{2}})$
Simplify Using Logarithmic Property: Now we can rewrite the original expression using the combined logarithm: $11^{\log_{11}(\sqrt{y}) - \log_{11}(z^{2})} = 11^{\log_{11}(\sqrt{y}/z^{2})}$
Final Simplified Expression: Using the property $a^{\log_a(b)} = b$ , we can simplify the expression to: $\newline$ $11^{\log_{11}(\sqrt{y}/z^{2})} = \sqrt{y}/z^{2}$
Final Simplified Expression: Using the property $a^{\log_a(b)} = b$ , we can simplify the expression to: $\newline$ $11^{\log_{11}(\sqrt{y}/z^{2})} = \sqrt{y}/z^{2}$ Since $\sqrt{y}$ is the square root of $y$ , we can write it as $y^{1/2}$ . So the final simplified expression is: $\newline$ $y^{1/2} / z^{2}$

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