(log(sqrtx-2))/(3y)-sqrt(64-x^(2)-y^(2))

Identify Properties: Identify the properties of logarithms and square roots to simplify the given expression (log(sqrt(x)-2))/(3y)-sqrt(64-x^2-y^2). For the logarithm part, we will use the property that allows us to express the logarithm of a power as a multiple of a logarithm. For the square root part, we will simplify the square root expression.Apply Logarithm Property: Apply the property of logarithms to the term (log(sqrt(x)-2))/(3y). The property is log_b(a^n) = n * log_b(a). Here, we have a square root, which is equivalent to the power of 1/2. So we can rewrite the logarithm as: (1/2 * log(x-2))/(3y).Simplify Square Root: Simplify the square root expression sqrt(64-x^2-y^2). We recognize that 64-x^2-y^2 could be a difference of squares if x^2 + y^2 equals 64. However, without additional information, we cannot simplify this further. We leave it as sqrt(64-x^2-y^2).Combine Results: Combine the results from Step 2 and Step 3 to write the final simplified expression. The final expression is (1/6y) * log(x-2) - sqrt(64-x^2-y^2).

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$\left(\frac{\log(\sqrt{x}-2)}{3y}\right)-\sqrt{64-x^{2}-y^{2}}$

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Algebra 2

Product property of logarithms

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\left(\frac{\log(\sqrt{x}-2)}{3y}\right)-\sqrt{64-x^{2}-y^{2}}

Identify Properties: Identify the properties of logarithms and square roots to simplify the given expression $(\log(\sqrt{x}-2))/(3y)-\sqrt{64-x^2-y^2}$ . For the logarithm part, we will use the property that allows us to express the logarithm of a power as a multiple of a logarithm. For the square root part, we will simplify the square root expression.
Apply Logarithm Property: Apply the property of logarithms to the term $(\log(\sqrt{x}-2))/(3y)$ . $\newline$ The property is $\log_b(a^n) = n \cdot \log_b(a)$ . Here, we have a square root, which is equivalent to the power of $1/2$ . So we can rewrite the logarithm as: $\newline$ $(1/2 \cdot \log(x-2))/(3y)$ .
Simplify Square Root: Simplify the square root expression $\sqrt{64-x^2-y^2}$ . We recognize that $64-x^2-y^2$ could be a difference of squares if $x^2 + y^2$ equals $64$ . However, without additional information, we cannot simplify this further. We leave it as $\sqrt{64-x^2-y^2}$ .
Combine Results: Combine the results from Step $2$ and Step $3$ to write the final simplified expression. $\newline$ The final expression is $(\frac{1}{6y}) \cdot \log(x-2) - \sqrt{64-x^2-y^2}$ .