Question

determine whether each expression is equivalent to $64^{5t^2 + 3t}$.
equivalent not equivalent
\\(\frac{128^{5t^2 - 3t}}{2^{5t^2 - 3t}}\\)
\\(\frac{32^{5t^2 + 3t}}{4^{5t^2 + 3t}}\\)
\\(2^{5t^2 + 3t} \cdot 4^{5t^2 + 3t}\\)

Explanation:

Step1: Simplify first expression

Rewrite bases as powers of 2:
$128=2^7$, $2=2^1$
$\frac{128^{5t^2-3t}}{2^{5t^2-3t}}=\frac{(2^7)^{5t^2-3t}}{2^{5t^2-3t}}=\frac{2^{7(5t^2-3t)}}{2^{5t^2-3t}}=2^{(7-1)(5t^2-3t)}=2^{6(5t^2-3t)}=(2^6)^{5t^2-3t}=64^{5t^2-3t}$
This does not match $64^{5t^2+3t}$.

Step2: Simplify second expression

Rewrite bases as powers of 2:
$32=2^5$, $4=2^2$
$\frac{32^{5t^2+3t}}{4^{5t^2+3t}}=\frac{(2^5)^{5t^2+3t}}{(2^2)^{5t^2+3t}}=\frac{2^{5(5t^2+3t)}}{2^{2(5t^2+3t)}}=2^{(5-2)(5t^2+3t)}=2^{3(5t^2+3t)}=(2^3)^{5t^2+3t}=8^{5t^2+3t}$
This does not match $64^{5t^2+3t}$.

Step3: Simplify third expression

Rewrite base 4 as power of 2:
$4=2^2$
$2^{5t^2+3t} \cdot 4^{5t^2+3t}=2^{5t^2+3t} \cdot (2^2)^{5t^2+3t}=2^{5t^2+3t} \cdot 2^{2(5t^2+3t)}=2^{(1+2)(5t^2+3t)}=2^{3(5t^2+3t)}$
Wait, correct rewrite: $64=2^6$, so:
$2^{5t^2+3t} \cdot 4^{5t^2+3t}=2^{5t^2+3t} \cdot (2^2)^{5t^2+3t}=2^{5t^2+3t + 2(5t^2+3t)}=2^{3(5t^2+3t)}$
Wait, error correction: $64=2^6$, so we need $2^{6(5t^2+3t)}$. Instead, $2 \cdot 4=2 \cdot 2^2=2^3$, so $(2 \cdot 4)^{5t^2+3t}=8^{5t^2+3t}$. No, wait:
$2^{A} \cdot 4^{A}=2^A \cdot (2^2)^A=2^A \cdot 2^{2A}=2^{3A}=8^A$, where $A=5t^2+3t$. This is not $64^A=(2^6)^A=2^{6A}$.

Wait correction for first expression:
$\frac{128^{B}}{2^{B}}=(128/2)^B=64^B$, where $B=5t^2-3t$. So it is $64^{5t^2-3t}$, not $64^{5t^2+3t}$.

Answer:

1. $\boldsymbol{\frac{128^{5t^2-3t}}{2^{5t^2-3t}}}$: Not Equivalent
2. $\boldsymbol{\frac{32^{5t^2+3t}}{4^{5t^2+3t}}}$: Not Equivalent
3. $\boldsymbol{2^{5t^2+3t} \cdot 4^{5t^2+3t}}$: Not Equivalent