To determine which of the provided options is equivalent to the quotient [tex]\(\frac{\sqrt{140}}{\sqrt{8}}\)[/tex], let's first simplify the expression step-by-step.
1. Combine the square roots:
[tex]\[
\frac{\sqrt{140}}{\sqrt{8}} = \sqrt{\frac{140}{8}}
\][/tex]
2. Simplify the fraction inside the square root:
[tex]\[
\frac{140}{8} \text{ simplifies to } \frac{140 \div 4}{8 \div 4} = \frac{35}{2}
\][/tex]
3. Rewrite the simplified expression:
[tex]\[
\frac{\sqrt{140}}{\sqrt{8}} = \sqrt{\frac{35}{2}}
\][/tex]
4. Rationalize the denominator (if needed):
We can split the square root of a fraction into the square root of the numerator and the denominator:
[tex]\[
\sqrt{\frac{35}{2}} = \frac{\sqrt{35}}{\sqrt{2}}
\][/tex]
5. Simplify further:
[tex]\[
\frac{\sqrt{35}}{\sqrt{2}} = \frac{\sqrt{35}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{35} \times \sqrt{2}}{2} = \frac{\sqrt{70}}{2}
\][/tex]
Thus, we have simplified the original expression [tex]\(\frac{\sqrt{140}}{\sqrt{8}}\)[/tex] to [tex]\(\frac{\sqrt{70}}{2}\)[/tex].
Given the provided options:
- A. [tex]\(\sqrt{70}\)[/tex]
- B. [tex]\(\frac{\sqrt{35}}{2}\)[/tex]
- C. [tex]\(\sqrt{35}\)[/tex]
- D. [tex]\(\frac{\sqrt{70}}{2}\)[/tex]
The correct option that matches our simplified expression [tex]\(\frac{\sqrt{70}}{2}\)[/tex] is D.
