The number $u$ is irrational. Which statement about $\sqrt{12} \cdot u$ is true? $\newline$ $\sqrt{12} \cdot u$ is rational. $\newline$ $\sqrt{12} \cdot u$ is irrational. $\newline$ $\sqrt{12} \cdot u$ can be rational or irrational, depending on the value of $u$ . $\newline$ Submit

Full solution

Q. The number

u

is irrational. Which statement about

\sqrt{12} \cdot u

is true?

\newline

\sqrt{12} \cdot u

is rational.

\newline

\sqrt{12} \cdot u

is irrational.

\newline

\sqrt{12} \cdot u

can be rational or irrational, depending on the value of

u

\newline

Submit

Factorize $12$ : Factorize $12$ to identify perfect squares. $\newline$ $12 = 2 \times 2 \times 3 = 2^2 \times 3$
Apply property of square roots: Apply the property of square roots over multiplication. $\sqrt{12u} = \sqrt{2^2 \times 3 \times u} = \sqrt{2^2} \times \sqrt{3u} = 2 \times \sqrt{3u}$
Analyze $\sqrt{3u}$ : Analyze the expression $\sqrt{3u}$ . $\newline$ Since $u$ is irrational, multiplying it by $3$ (a rational number) keeps it irrational (property of irrationals).
Determine nature of $\sqrt{3*u}$ : Determine the nature of $\sqrt{3*u}$ . The square root of an irrational number is irrational.
Conclude nature of $\sqrt{12u}$ : Conclude the nature of $\sqrt{12u}$ . $\newline$ Since $\sqrt{3u}$ is irrational, multiplying it by $2$ (a rational number) still results in an irrational number.