Which expression is equivalent to x Superscript StartFraction 2 over 3 EndFraction Baseline plus StartFraction 1 over x Superscript StartFraction 2 over 3 EndFraction Baseline EndFraction?

Question

Answer options with 5 options
A.
StartFraction x Superscript StartFraction 2 over 3 EndFraction Baseline plus 1 over x Superscript StartFraction 2 over 3 EndFraction Baseline EndFraction

B.
StartFraction x Superscript StartFraction 4 over 9 EndFraction Baseline plus 1 over x Superscript StartFraction 2 over 3 EndFraction Baseline EndFraction

C.
1

D.
StartFraction x plus 1 over x Superscript StartFraction 2 over 3 EndFraction Baseline EndFraction

E.
StartFraction x Superscript StartFraction 4 over 3 EndFraction Baseline plus 1 over x Superscript StartFraction 2 over 3 EndFraction Baseline EndFraction

Answer 1

We want to simplify the expression \( x^{\frac{2}{3}} + \frac{1}{x^{\frac{2}{3}}} \).

This expression can also be thought of as a single fraction:

\[ x^{\frac{2}{3}} + \frac{1}{x^{\frac{2}{3}}} = \frac{x^{\frac{2}{3}} \cdot x^{\frac{2}{3}} + 1}{x^{\frac{2}{3}}} \]

This gives us:

\[ = \frac{x^{\frac{4}{3}} + 1}{x^{\frac{2}{3}}} \]

Now, let's analyze the options provided:

A. \( \frac{x^{\frac{2}{3}} + 1}{x^{\frac{2}{3}}} \) - This does not match our expression.

B. \( \frac{x^{\frac{4}{9}} + 1}{x^{\frac{2}{3}}} \) - The exponent on x is incorrect.

C. \( 1 \) - This does not simplify the expression we have.

D. \( \frac{x + 1}{x^{\frac{2}{3}}} \) - This is also incorrect as the numerator does not equal \( x^{\frac{4}{3}} + 1 \).

E. \( \frac{x^{\frac{4}{3}} + 1}{x^{\frac{2}{3}}} \) - This matches our simplified expression exactly.

Thus, the correct option is:

E. \( \frac{x^{\frac{4}{3}} + 1}{x^{\frac{2}{3}}} \).