To determine which choice is equivalent to the quotient [tex]\(\sqrt{30(x-1)} \div \sqrt{5(x-1)^2}\)[/tex] for acceptable values of [tex]\(x\)[/tex], we'll simplify the expression step by step.
First, let's rewrite the given quotient using division of square roots:
[tex]\[
\frac{\sqrt{30(x-1)}}{\sqrt{5(x-1)^2}}
\][/tex]
Next, use the property of square roots to combine the numerator and denominator:
[tex]\[
\sqrt{\frac{30(x-1)}{5(x-1)^2}}
\][/tex]
Now, simplify the expression under the square root:
[tex]\[
\sqrt{\frac{30(x-1)}{5(x-1)^2}} = \sqrt{\frac{30}{5} \cdot \frac{(x-1)}{(x-1)^2}}
\][/tex]
Simplify the constants and the powers of [tex]\(x-1\)[/tex]:
[tex]\[
\sqrt{6 \cdot \frac{1}{x-1}} = \sqrt{\frac{6}{x-1}}
\][/tex]
Thus, the simplified form of the given quotient is:
[tex]\[
\sqrt{\frac{6}{x-1}}
\][/tex]
Finally, compare this result with the given choices:
- A. [tex]\(\sqrt{150(x-1)^3}\)[/tex]
- B. [tex]\(\sqrt{6(x-1)}\)[/tex]
- C. [tex]\(\sqrt{30(x-1) - 5(x-1)^2}\)[/tex]
- D. [tex]\(\sqrt{\frac{6}{(x-1)}}\)[/tex]
The correct answer is:
D. [tex]\(\sqrt{\frac{6}{(x-1)}}\)[/tex]
