Select the correct answer.

Which of the following is equal to the expression below?

[tex]\[ (160 \cdot 243)^{\frac{2}{6}} \][/tex]

A. [tex]\(5 \sqrt[3]{5}\)[/tex]

B. 96

C. 80

D. [tex]\(6 \sqrt[3]{5}\)[/tex]



Answer :

Sure, let's work through the question step-by-step.

We need to simplify the expression [tex]\((160 \cdot 243)^{\frac{2}{6}}\)[/tex].

1. Calculate the product of the numbers inside the parenthesis:

[tex]\[ 160 \cdot 243 \][/tex]
Which equals:

[tex]\[ 160 \cdot 243 = 38880 \][/tex]

2. Simplify the exponent:

The exponent [tex]\(\frac{2}{6}\)[/tex] can be simplified to [tex]\(\frac{1}{3}\)[/tex].

So the expression becomes:

[tex]\[ 38880^{\frac{1}{3}} \][/tex]

3. Calculate the cube root:

The cube root of 38880 is approximately:

[tex]\[ 38880^{\frac{1}{3}} \approx 33.87729703971702 \][/tex]

Now we need to match this number, [tex]\(33.87729703971702\)[/tex], with one of the given options:

A. [tex]\(5 \sqrt[3]{5}\)[/tex]
B. 96
C. 80
D. [tex]\(6 \sqrt[3]{5}\)[/tex]

Let's determine the numerical value for each option.

A. [tex]\(5 \sqrt[3]{5}\)[/tex]:

[tex]\[ 5 \cdot \sqrt[3]{5} \approx 5 \cdot 1.710 \approx 8.55 \quad (\text{not close}) \][/tex]

B. 96:

[tex]\[ (\text{exact number, no calculation needed, not close}) \][/tex]

C. 80:

[tex]\[ (\text{exact number, no calculation needed, not close}) \][/tex]

D. [tex]\(6 \sqrt[3]{5}\)[/tex]:

[tex]\[ 6 \cdot \sqrt[3]{5} \approx 6 \cdot 1.710 \approx 10.26 \quad (\text{not close}) \][/tex]

None of the options given exactly match [tex]\(33.87729703971702\)[/tex], however, choice D ([tex]\(6 \sqrt[3]{5}\)[/tex]) is the closest approximation among them, but it still does not match the calculated value. Probably there is a problem in the options given.