To determine whether the expression \( \frac{x + 1}{\sqrt{x + 1}} \) is a whole number for the given values of \( x \), we can simplify the expression first:
\[ \frac{x + 1}{\sqrt{x + 1}} = \sqrt{x + 1} \cdot \frac{x + 1}{x + 1} = \sqrt{x + 1} \text{ (assuming } x + 1 \neq 0\text{)} \]
This means we need \( \sqrt{x + 1} \) to be a whole number, or equivalently, \( x + 1 \) must be a perfect square. Let's evaluate the provided values of \( x \):
-
For \( x = 5 \): \[ x + 1 = 5 + 1 = 6 \quad (\text{not a perfect square}) \ \frac{5 + 1}{\sqrt{5 + 1}} = \frac{6}{\sqrt{6}} \text{ (not a whole number)} \] False
-
For \( x = 0 \): \[ x + 1 = 0 + 1 = 1 \quad (\text{perfect square}) \ \frac{0 + 1}{\sqrt{0 + 1}} = \frac{1}{1} = 1 \text{ (whole number)} \] True
-
For \( x = 15 \): \[ x + 1 = 15 + 1 = 16 \quad (\text{perfect square}) \ \frac{15 + 1}{\sqrt{15 + 1}} = \frac{16}{4} = 4 \text{ (whole number)} \] True
-
For \( x = -5 \): \[ x + 1 = -5 + 1 = -4 \quad (\text{not a perfect square}) \ \frac{-5 + 1}{\sqrt{-5 + 1}} = \frac{-4}{\sqrt{-4}} \text{ (undefined)} \] False
To summarize:
- \( x = 5 \): False
- \( x = 0 \): True
- \( x = 15 \): True
- \( x = -5 \): False
2 answers