arrow_back Given \[ y=\sqrt{x^{7}}-\frac{5}{x^{3}} \] Which of the following is equivalent to \(y\) ?

Question

Given
\[
y=\sqrt{x^{7}}-\frac{5}{x^{3}}
\]
Which of the following is equivalent to \(y\) ?

\begin{align}
\begin{array}{|l|l|}
\hline \text { A } & x^{14}-5 x^{-3} \\
\hline \text { B } & x^{\frac{7}{2}}+5 x^{-3} \\
\hline \text { C } & x^{\frac{7}{2}}-5 x^{-3} \\
\hline \text { D } & x^{\frac{7}{2}}-5 x^{\frac{1}{3}} \\
\hline
\end{array}
\end{align}

Answer 1

Identify the correct expression for \(y\)
Looking at the first term:
\[
\begin{aligned}
y_{1} &=\sqrt{x^{7}} \\
&=\left(x^{7}\right)^{\frac{1}{2}} \\
&=x^{\frac{7}{2}}
\end{aligned}
\]
And then looking at the second term:
\[
\begin{aligned}
y_{2} &=-\frac{5}{x^{3}} \\
&=-5 x^{-3}
\end{aligned}
\]
Putting these together gives us:
\[
\begin{aligned}
y &=y_{1}+y_{2} \\
&=x^{\frac{7}{2}}-5 x^{-3}
\end{aligned}
\]
So the correct choice is Option \(\mathbf{C}\).