To determine which of the given expressions are equivalent to [tex]\(\left(\frac{750}{512}\right)^{\frac{1}{3}}\)[/tex], we need to evaluate each one numerically and compare it to the value of the given expression.
Given:
[tex]\[
\left(\frac{750}{512}\right)^{\frac{1}{3}} \approx 1.1357003705200872
\][/tex]
Let's evaluate each provided option:
Option 1: [tex]\(\frac{5}{8}\)[/tex]
[tex]\[
\frac{5}{8} = 0.625
\][/tex]
Option 2: [tex]\(\frac{\pi 30}{150}\)[/tex]
[tex]\[
\frac{\pi \cdot 30}{150} \approx 0.6283185307179586
\][/tex]
Option 3: [tex]\(\sqrt[4]{\frac{70}{612}}\)[/tex]
[tex]\[
\left(\frac{70}{612}\right)^{\frac{1}{4}} \approx 0.5815494568798613
\][/tex]
Option 4: [tex]\(\frac{1750}{512}\)[/tex]
[tex]\[
\frac{1750}{512} \approx 3.41796875
\][/tex]
Option 5: [tex]\(\frac{5}{8} \sqrt[3]{6}\)[/tex]
[tex]\[
\frac{5}{8} \cdot 6^{\frac{1}{3}} \approx 1.1357003705200872
\][/tex]
Option 6: [tex]\(\frac{75}{\sqrt{812}}\)[/tex]
[tex]\[
\frac{75}{\sqrt{812}} \approx 2.631984023788487
\][/tex]
Comparing all these values to [tex]\(\left(\frac{750}{512}\right)^{\frac{1}{3}}\)[/tex]:
[tex]\[
\left(\frac{750}{512}\right)^{\frac{1}{3}} \approx 1.1357003705200872
\][/tex]
Correct matches:
- Option 5: [tex]\(\frac{5}{8} \sqrt[3]{6} \approx 1.1357003705200872\)[/tex]
Therefore, the only correct answer is:
[tex]\[
\frac{5}{8} \sqrt[3]{6}
\][/tex]
