Problem: The symbol ∣a∣|a|∣a∣ means aaa if aaa is a positive number or zero, and −a-a−a if aaa is a negative number. For all real values of ttt the expression t4+t2\sqrt{t^{4}+t^{2}}t4+t2 is equal to:
Answer Choices:
A. t3\mathrm{t}^{3}t3
B. t2+t\mathrm{t}^{2}+\mathrm{t}t2+t
C. ∣t2+t∣\left|t^{2}+t\right|∣∣∣t2+t∣∣∣
D. tt2+1t \sqrt{t^{2}+1}tt2+1
E. ∣t∣1+t2|\mathrm{t}| \sqrt{1+\mathrm{t}^{2}}∣t∣1+t2 Solution:
t4+t3=t2(t2+1)=t2t2+1=∣t∣1+t2\sqrt{t^{4}+t^{3}}=\sqrt{t^{2}\left(t^{2}+1\right)}=\sqrt{t^{2}} \sqrt{t^{2}+1}= \mid t \mid \sqrt{1+t^{2}}t4+t3=t2(t2+1)=t2t2+1=∣t∣1+t2