# 1961 AHSME Problems/Problem 13

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## Problem

The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to?

$\textbf{(A)}\ t^3\qquad \textbf{(B)}\ t^2+t\qquad \textbf{(C)}\ |t^2+t|\qquad \textbf{(D)}\ t\sqrt{t^2+1}\qquad \textbf{(E)}\ |t|\sqrt{1+t^2}$

## Solution

Factor out the $t^2$ inside the square root. $$\sqrt{t^2 \cdot (t^2 + 1)}$$ $$\sqrt{t^2} \cdot \sqrt{t^2 + 1}$$ Remember that $\sqrt{t^2} = |t|$ because square rooting a nonnegative real number will always result in a nonnegative number. $$|t|\sqrt{t^2 + 1}$$ The answer is $\boxed{\textbf{(E)}}$.

 1961 AHSC (Problems • Answer Key • Resources) Preceded byProblem 12 Followed byProblem 14 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 All AHSME Problems and Solutions