Question

which expression is equivalent to $\frac{8^{4}}{2^{4}}$? $\frac{1}{4^{4}}$ $4^{-1}cdot4^{5}$ $4cdot4^{4}$ $\frac{4^{8}}{16}$

Explanation:

Step1: Simplify $\frac{8^{4}}{2^{4}}$

Use the power - of - a - quotient rule $\frac{a^{n}}{b^{n}}=(\frac{a}{b})^{n}$. Here $a = 8$, $b = 2$ and $n = 4$, so $\frac{8^{4}}{2^{4}}=(\frac{8}{2})^{4}=4^{4}$.

Step2: Simplify each option

• Option 1: $\frac{1}{4^{4}}$ is not equal to $4^{4}$.
• Option 2: Use the rule $a^{m}\cdot a^{n}=a^{m + n}$. For $4^{-1}\cdot4^{5}$, we have $4^{-1 + 5}=4^{4}$.
• Option 3: $4\cdot4^{4}=4^{1+4}=4^{5}

eq4^{4}$.

• Option 4: $\frac{4^{8}}{16}=\frac{4^{8}}{4^{2}}$. Using the rule $\frac{a^{m}}{a^{n}}=a^{m - n}$, we get $4^{8 - 2}=4^{6}

eq4^{4}$.

Answer:

$4^{-1}\cdot4^{5}$