Question

error analysis which of the following describes the error in simplifying the expression?$sqrt6{\frac{64h^{12}}{g^{6}}}=\frac{sqrt6{64h^{12}}}{sqrt6{g^{6}}}LXB0=\frac{2h^{2}}{g}$○ the product property of radicals was applied incorrectly.○ the quotient property of radicals was applied incorrectly.○ absolute value symbols are needed around one of the variables in the simplified expression.○ absolute value symbols are needed around both variables in the simplified expression.correct the error.the simplest form of the expression is $\frac{square}{square}$.

Explanation:

Step1: Analyze radical simplification rules

For even-indexed radicals ($\sqrt[n]{x}$ where $n$ is even), $\sqrt[n]{x^n} = |x|$ to ensure a non-negative result. Here, we have a 6th root (even index).

Step2: Check the final step error

When simplifying $\frac{\sqrt[6]{2^6 \cdot (h^2)^6}}{\sqrt[6]{g^6}}$, $\sqrt[6]{g^6} = |g|$, while $\sqrt[6]{(h^2)^6}=h^2$ (since $h^2$ is non-negative for all real $h$) and $\sqrt[6]{2^6}=2$. The original work omitted the absolute value for $g$.

Step3: Correct the expression

Apply the even-index radical rule to $g$:
$\frac{\sqrt[6]{64h^{12}}}{\sqrt[6]{g^6}} = \frac{2h^2}{|g|}$

Answer:

Error Identification:

Absolute value symbols are needed around one of the variables in the simplified expression.

Corrected Simplest Form:

$\frac{2h^2}{|g|}$