Question

select all the expressions that are equivalent to $4^4 \cdot 4^3$.
$\frac{4^9}{4^2}$ $(4^7)^1$
$2^7 \cdot 2^7$ $(4^6)^1$

Explanation:

Step1: Simplify original expression

$4^4 \cdot 4^3 = 4^{4+3} = 4^7$

Step2: Check first option

$\frac{4^9}{4^2} = 4^{9-2} = 4^7$ (equivalent)

Step3: Check second option

$(4^7)^1 = 4^{7 \cdot 1} = 4^7$ (equivalent)

Step4: Check third option

$2^7 \cdot 2^7 = 2^{14} = (2^2)^7 = 4^7$? Wait, $2^{14}=16384$, $4^7=16384$? Wait no: $2^7 \cdot 2^7=2^{14}=(2^2)^7=4^7$, but wait original is $4^7$. Wait wait, $2^7 \cdot 2^7=2^{14}=16384$, $4^7=16384$. But wait let's recheck: $4^7=(2^2)^7=2^{14}$, yes. But wait the fourth option: $(4^6)^1=4^6≠4^7$. Wait wait the third option: $2^7 \cdot 2^7=2^{14}=4^7$, yes. But wait the problem says "select all". Wait wait my mistake earlier: $2^7 \cdot 2^7=2^{14}=4^7$, so it is equivalent. Wait wait let's confirm: $2^7=128$, $128*128=16384$; $4^7=16384$. Yes. So why did I think no? Oh no, I made a mistake. Wait the third option is equivalent. Wait the fourth option is $4^6≠4^7$. So the equivalent ones are first, second, third.

Wait correction:

Step4: Check third option

$2^7 \cdot 2^7 = 2^{14} = (2^2)^7 = 4^7$ (equivalent)

Step5: Check fourth option

$(4^6)^1 = 4^6 ≠ 4^7$ (not equivalent)

So the answer is $\frac{4^9}{4^2}$, $(4^7)^1$, $2^7 \cdot 2^7$

Step1: Simplify original expression

$4^4 \cdot 4^3 = 4^{7}$

Step2: Evaluate first option

$\frac{4^9}{4^2} = 4^{7}$ (equivalent)

Step3: Evaluate second option

$(4^7)^1 = 4^{7}$ (equivalent)

Step4: Evaluate third option

$2^7 \cdot 2^7 = 2^{14} = 4^{7}$ (equivalent)

Step5: Evaluate fourth option

$(4^6)^1 = 4^6 ≠ 4^7$ (not equivalent)

Answer:

$\frac{4^9}{4^2}$, $(4^7)^1$