Exercise 42

The expression \( \cfrac{x - 5}{\sqrt{x - 5}} \) equals:

  1. \( 1 \)

  2. \( \cfrac{ \sqrt{ (x - 5)^3 } }{x - 5} \)

  3. \( \cfrac{2x}{x - 5} \)

  4. \( \cfrac{x + 5}{x - 5} \)

  5. \( \cfrac{x^2 - 25}{x - 5} \)

Try to solve it before checking the answer.

  1. \( \cfrac{ \sqrt{ (x – 5)^3 } }{x – 5} \)

Let’s rationalize the denominator by multiplying and dividing by \( \sqrt{ x – 5 } \):

\[ \begin{aligned} &\frac{x – 5}{ \sqrt{x – 5} } \times \frac{ \sqrt{x – 5} }{\sqrt{x – 5}} \\[.5em] &\hspace{7em}= \frac{ (x – 5) \sqrt{ x – 5 } }{ \sqrt{x – 5}{ \sqrt{ x – 5 } } } \\[.5em] &\hspace{7em}= \frac{ \sqrt{ (x – 5)^2 (x – 5) } }{x – 5} \\[.5em] &\hspace{7em}= \boldsymbol{ \frac{ \sqrt{ (x – 5)^3 } }{x – 5} } \end{aligned} \]
\[ \begin{aligned} \frac{x – 5}{ \sqrt{x – 5} } \times \frac{ \sqrt{x – 5} }{\sqrt{x – 5}} &= \frac{ (x – 5) \sqrt{ x – 5 } }{ \sqrt{x – 5}{ \sqrt{ x – 5 } } } \\[.5em] &= \frac{ \sqrt{ (x – 5)^2 (x – 5) } }{x – 5} \\[.5em] &= \boldsymbol{ \frac{ \sqrt{ (x – 5)^3 } }{x – 5} } \end{aligned} \]

More Math Reasoning Exercises