Question

determine whether each expression is equivalent to $3^{t^2 + 3t}$.
equivalent not equivalent
$9^t \cdot 27^t$
$(3^t)^{t + 3}$
$3^{t^2} \cdot 27^t$
show calculator

Explanation:

Step1: Rewrite bases as powers of 3

$9^t = (3^2)^t = 3^{2t}$, $27^t=(3^3)^t=3^{3t}$

Step2: Simplify first expression

$9^t \cdot 27^t = 3^{2t} \cdot 3^{3t} = 3^{2t+3t}=3^{5t}$

Step3: Simplify second expression

$(3^t)^{t+3}=3^{t(t+3)}=3^{t^2+3t}$

Step4: Simplify third expression

$27^t=3^{3t}$, so $3^{t^2} \cdot 27^t=3^{t^2} \cdot 3^{3t}=3^{t^2+3t}$

Answer:

• $9^t \cdot 27^t$: Not Equivalent
• $(3^t)^{t+3}$: Equivalent
• $3^{t^2} \cdot 27^t$: Equivalent