Let's analyze each of the given options to determine which expression, if any, is equivalent to [tex]\(2x^{\frac{2}{3}} + x \sqrt[3]{16}\)[/tex].
Consider the original expression:
[tex]\[ 2x^{\frac{2}{3}} + x \sqrt[3]{16} \][/tex]
Option 1: [tex]\( 2 \sqrt[3]{x^2} + 4^{\frac{2}{3}} x \)[/tex]
We need to consider the individual terms:
[tex]\[ 2 \sqrt[3]{x^2} = 2 x^{\frac{2}{3}} \][/tex]
[tex]\[ 4^{\frac{2}{3}} x \][/tex]
Now let's simplify [tex]\( 4^{\frac{2}{3}} \)[/tex]:
[tex]\[ 4^{\frac{2}{3}} = (2^2)^{\frac{2}{3}} = 2^{\frac{4}{3}} \][/tex]
So, the expression becomes:
[tex]\[ 2 x^{\frac{2}{3}} + 2^{\frac{4}{3}} x \][/tex]
This doesn't match [tex]\( 2x^{\frac{2}{3}} + x \sqrt[3]{16} \)[/tex] because [tex]\( x \sqrt[3]{16} \)[/tex] simplifies to [tex]\( x \cdot 2 \cdot \sqrt[3]{2} \)[/tex], not [tex]\( 2^{4/3}x \)[/tex].
Option 2: [tex]\((2x)^{\frac{2}{3}} + 16^{\frac{1}{3}} x\)[/tex]
First, let's simplify each term:
[tex]\[ (2x)^{\frac{2}{3}} = 2^{\frac{2}{3}}x^{\frac{2}{3}} \][/tex]
[tex]\[ 16^{\frac{1}{3}} x = 2 x \][/tex]
So, the expression becomes:
[tex]\[ 2^{\frac{2}{3}}x^{\frac{2}{3}} + 2x \][/tex]
Again, this does not match the original expression [tex]\( 2x^{\frac{2}{3}} + x \sqrt[3]{16} \)[/tex].
Option 3: [tex]\( 2 (\sqrt[3]{x})^2 + 4x \)[/tex]
Simplify each term:
[tex]\[ 2 (\sqrt[3]{x})^2 = 2 (x^{\frac{1}{3}})^2 = 2 x^{\frac{2}{3}} \][/tex]
[tex]\[ 4x = 4x \][/tex]
So, the expression becomes:
[tex]\[ 2 x^{\frac{2}{3}} + 4x \][/tex]
This does not match [tex]\( 2x^{\frac{2}{3}} + x \sqrt[3]{16} \)[/tex].
Option 4: [tex]\(\sqrt[3]{2x^2} + (16x)^{\frac{1}{3}}\)[/tex]
Simplify each term:
[tex]\[ \sqrt[3]{2x^2} = (2x^2)^{\frac{1}{3}} = 2^{\frac{1}{3}} x^{\frac{2}{3}} \][/tex]
[tex]\[ (16x)^{\frac{1}{3}} = 16^{\frac{1}{3}} x^{\frac{1}{3}} = 2 x^{\frac{1}{3}} \][/tex]
So, the expression becomes:
[tex]\[ 2^{\frac{1}{3}} x^{\frac{2}{3}} + 2x^{\frac{1}{3}} \][/tex]
This also does not match [tex]\( 2x^{\frac{2}{3}} + x \sqrt[3]{16} \)[/tex].
After examining all the options, none of them are equivalent to [tex]\(2x^{\frac{2}{3}} + x \sqrt[3]{16}\)[/tex]. Therefore, the correct answer is that none of the options match the original expression.
