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The number $c$ is rational. Which statement about $\sqrt{7} - c$ is true? $\newline$ Choices: $\newline$ (A) $\sqrt{7} - c$ is rational. $\newline$ (B) $\sqrt{7} - c$ is irrational. $\newline$ (C) $\sqrt{7} - c$ can be rational or irrational, depending on the value of $c$ .

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Properties of operations on rational and irrational numbers

Full solution

Q. The number

c

is rational. Which statement about

\sqrt{7} - c

is true?

\newline

Choices:

\newline

(A)

\sqrt{7} - c

is rational.

\newline

(B)

\sqrt{7} - c

is irrational.

\newline

(C)

\sqrt{7} - c

can be rational or irrational, depending on the value of

c

.

Identify Type of $\sqrt{7}$ : Identify whether $\sqrt{7}$ is a rational or irrational number. $7$ is a non-perfect square, which means that $\sqrt{7}$ is an irrational number.
Consider Nature of Number $c$ : Consider the nature of the number $c$ . Since $c$ is given as a rational number, we know that it can be expressed as a fraction of two integers, where the denominator is not zero.
Analyze $\sqrt{7} - c$ : Analyze the expression $\sqrt{7} - c$ . Since $\sqrt{7}$ is irrational and $c$ is rational, subtracting a rational number from an irrational number will result in an irrational number. This is because you cannot express an irrational number as a fraction of two integers, and subtracting a fraction (rational number) from it will not change this fact.
Conclude Result: Conclude the nature of $\sqrt{7} - c$ . Based on the previous steps, $\sqrt{7} - c$ is always irrational, regardless of the value of the rational number $c$ .

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