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 $\boxed{\textbf{(E)}}$.

See Also

1961 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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