Simplify the expression. $\frac { t - 4 } { t ^ { 2 } - 16 } \cdot \frac { t + 4 } { t ^ { 2 } + 16 }$
A) $\frac { 1 } { t - 16 }$
B) $\frac { 1 } { t ^ { 2 } + 16 }$
C) $t ^ { 2 } + 16$
D) $\frac { t ^ { 2 } - 4 } { t + 16 }$
E) $\frac { t ^ { 2 } + 4 } { t + 16 }$

Question

Quizplus · Accepted Answer

To simplify the given expression, we can cancel out the common factor of $t-4$ and $t+4$ in the numerator and denominator respectively. This gives us: $$\frac{t-4}{t^2-16}\cdot\frac{t+4}{t^2+16}=\frac{1}{t^2-16}\cdot\frac{t+4}{t^2+16}.$$ Now, we can cancel out the common factor of $(t^2-16)$ in the numerator and denominator of the first fraction, and the common factor of $(t^2+16)$ in the numerator and denominator of the second fraction: $$\frac{1}{t^2-16}\cdot\frac{t+4}{t^2+16}=\frac{1}{(t+4)(t-4)}\cdot\frac{t+4}{t^2+16}.$$ Finally, we can cancel out the common factor of $(t+4)$ in the numerator and denominator, which gives us: $$\frac{1}{(t+4)(t-4)}\cdot\frac{t+4}{t^2+16}=\frac{1}{t^2-16} \cdot \frac{1}{1+\frac{16}{t^2}}=\frac{1}{t^2-16} \cdot \frac{1}{1+(\frac{4}{t})^2}.$$ Since the last expression is in its simplest form and cannot be further simplified, the answer is $\boxed{	extbf{(B)}}.$

Simplify the Expression t−4t2−16⋅t+4t2+16\frac { t - 4 } { t ^ { 2 } - 16 } \cdot \frac { t + 4 } { t ^ { 2 } + 16 }t2−16t−4​⋅t2+16t+4​

Simplify the Expression $\frac { t - 4 } { t ^ { 2 } - 16 } \cdot \frac { t + 4 } { t ^ { 2 } + 16 }$