The symbol |a| means a if a is a positive number or zero, and -a if a is a negative number. For all real values of t the expression √t^4+t^2 is equal to:

1961 AMC 12 Problem 13

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13.

The symbol

|a|

means

a

a

is a positive number or zero, and

-a

a

is a negative number. For all real values of

t

the expression

\sqrt{t^4+t^2}

is equal to:

t^3

t^2+t

|t^2+t|

t\sqrt{t^2+1}

|t|\sqrt{1+t^2}

Factor

t^2

from under the radical

For real

t,

\sqrt{t^2}=|t|

Factoring inside the radical gives

\begin{aligned} \sqrt{t^2(t^2+1)} &=\sqrt{t^2}\sqrt{t^2+1}\\ &=|t|\sqrt{1+t^2}. \end{aligned}

Therefore, the correct answer is E.