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Which values make square root X plus one a whole number

X = 5 true or false
X = 0 true or false
X = 15 true or false
X = -5 true or false

3 answers

To determine which values make \( \sqrt{X} + 1 \) a whole number, we can rewrite the expression as:

\[ \sqrt{X} + 1 = n \]

Where \( n \) is a whole number. This implies:

\[ \sqrt{X} = n - 1 \]

For \( \sqrt{X} \) to be a whole number, \( X \) must be a perfect square, which means \( X \) must be non-negative.

Now let's evaluate each provided value of \( X \):

  1. \( X = 5 \): \[ \sqrt{5} + 1 \text{ is not a whole number (since } \sqrt{5} \text{ is not an integer).} \] False

  2. \( X = 0 \): \[ \sqrt{0} + 1 = 0 + 1 = 1 \text{ (a whole number).} \] True

  3. \( X = 15 \): \[ \sqrt{15} + 1 \text{ is not a whole number (since } \sqrt{15} \text{ is not an integer).} \] False

  4. \( X = -5 \): \[ \sqrt{-5} \text{ is not a real number (it is imaginary).} \] False

To summarize:

  • \( X = 5 \): False
  • \( X = 0 \): True
  • \( X = 15 \): False
  • \( X = -5 \): False
Write the following infinite decimal expansion as a fraction you do not need to simplify 0.85

To write the infinite decimal expansion \(0.85\) as a fraction, we first note that \(0.85\) has a finite number of decimal places. It can be expressed simply as follows:

\[ 0.85 = \frac{85}{100} \]

This fraction is \( \frac{85}{100} \). Since you mentioned that you do not need to simplify, this is the answer. However, if you wanted to express it as an infinite decimal expansion such as \(0.850000...\), it would still represent the same value.

Thus, the answer can be stated as:

\[ \frac{85}{100} \]