The number y is rational. Which statement about sqrt 21 + y is true? Choices: (A)sqrt 21 + y is rational. (B)sqrt 21 + y is irrational. (C)sqrt 21 + y can be rational or irrational, depending on the value of y.

Question

The number y is rational. Which statement about sqrt 21 + y is true?   Choices: (A)sqrt 21 + y is rational. (B)sqrt 21 + y is irrational. (C)sqrt 21 + y can be rational or irrational, depending on the value of y.

Bytelearn · Accepted Answer

Identify Type of Number: Identify whether sqrt(21) is a rational or irrational number. 21 is a non-perfect square, which means that its square root cannot be expressed as a ratio of two integers. Therefore, sqrt(21) is an irrational number.Nature of Sum: Consider the nature of the sum of an irrational number and a rational number. By definition, a rational number can be expressed as a ratio of two integers, while an irrational number cannot. The sum of a rational number and an irrational number is always irrational. This is because if it were rational, then subtracting the rational number from both sides would leave an irrational number equal to a rational number, which is a contradiction.Apply Reasoning: Apply the above reasoning to sqrt(21) and y, where y is rational. Since sqrt(21) is irrational and y is rational, their sum, sqrt(21) + y, must be irrational.

The number $y$ is rational. Which statement about $\sqrt{21} + y$ is true? $\newline$ Choices: $\newline$ (A) $\sqrt{21} + y$ is rational. $\newline$ (B) $\sqrt{21} + y$ is irrational. $\newline$ (C) $\sqrt{21} + y$ can be rational or irrational, depending on the value of $y$ .

Full solution

More problems from Properties of operations on rational and irrational numbers

Related topics

The number yyy is rational. Which statement about 21+y\sqrt{21} + y21​+y is true?\newlineChoices:\newline(A) 21+y\sqrt{21} + y21​+y is rational.\newline(B) 21+y\sqrt{21} + y21​+y is irrational.\newline(C) 21+y\sqrt{21} + y21​+y can be rational or irrational, depending on the value of yyy.

The number $y$ is rational. Which statement about $\sqrt{21} + y$ is true? $\newline$ Choices: $\newline$ (A) $\sqrt{21} + y$ is rational. $\newline$ (B) $\sqrt{21} + y$ is irrational. $\newline$ (C) $\sqrt{21} + y$ can be rational or irrational, depending on the value of $y$ .