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Given [tex]\( x \ \textgreater \ 0 \)[/tex] and [tex]\( y \ \textgreater \ 0 \)[/tex], select the expression that is equivalent to

[tex]\[ \sqrt[3]{27 x^5 y^8} \][/tex]

A. [tex]\( 3 x^{\frac{5}{3}} y^{\frac{8}{3}} \)[/tex]
B. [tex]\( 9 x^{\frac{5}{3}} y^{\frac{8}{3}} \)[/tex]
C. [tex]\( 3 x^{\frac{3}{3}} y^{\frac{3}{8}} \)[/tex]
D. [tex]\( 9 x^3 y^{\frac{3}{8}} \)[/tex]



Answer :

GinnyAnswer
To determine which expression is equivalent to [tex]\(\sqrt[3]{27 x^5 y^8}\)[/tex], let's break down and simplify the given expression step-by-step.

Given expression: [tex]\(\sqrt[3]{27 x^5 y^8}\)[/tex]

Step 1: Break the expression under the cube root into separate parts that we can apply the cube root to individually:

[tex]\[ \sqrt[3]{27 x^5 y^8} = \sqrt[3]{27} \cdot \sqrt[3]{x^5} \cdot \sqrt[3]{y^8} \][/tex]

Step 2: Simplify each part individually:

1. [tex]\(\sqrt[3]{27} = 3\)[/tex]
2. [tex]\(\sqrt[3]{x^5} = x^{5/3}\)[/tex]
3. [tex]\(\sqrt[3]{y^8} = y^{8/3}\)[/tex]

Thus, the expression becomes:

[tex]\[ 3 \cdot x^{5/3} \cdot y^{8/3} \][/tex]

Step 3: Combine all the simplified parts into one expression:

[tex]\[ 3 x^{5/3} y^{8/3} \][/tex]

Now, compare this resulting expression with the given options:

1. [tex]\(3 x^{\frac{5}{5}} y^{\frac{8}{3}}\)[/tex]
2. [tex]\(9 x^{\frac{5}{3}} y^{\frac{8}{3}}\)[/tex]
3. [tex]\(3 x^{\frac{3}{3}} y^{\frac{3}{8}}\)[/tex]
4. [tex]\(9 x^3 y^{\frac{3}{8}}\)[/tex]

The correct expression [tex]\(3 x^{5/3} y^{8/3}\)[/tex] matches the first option:

Answer: [tex]\(3 x^{\frac{5}{3}} y^{\frac{8}{3}}\)[/tex]

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