Select all the expressions that are equivalent to the expression below:

[tex]\[
\left(\frac{750}{512}\right)^{\frac{1}{3}}
\][/tex]

A. [tex]\(\frac{5}{8}\)[/tex]

B. [tex]\(\frac{\pi 30}{150}\)[/tex]

C. [tex]\(\sqrt[4]{\frac{70}{612}}\)[/tex]

D. [tex]\(\frac{1750}{512}\)[/tex]

E. [tex]\(\frac{5}{8} \sqrt[3]{6}\)[/tex]

F. [tex]\(\frac{y 75}{\sqrt{812}}\)[/tex]



Answer :

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]