Write the Fraction in Simplest Form (4^(2x+1)*2^(1-x))/(8^(x+2))

$\frac{4^{2 x + 1} \cdot 2^{1 - x}}{8^{x + 2}}$

Step 1

Rewrite $4$ as $2^{2}$ .

$\frac{{(2^{2})}^{2 x + 1} \cdot 2^{1 - x}}{8^{x + 2}}$

Step 2

Multiply the exponents in ${(2^{2})}^{2 x + 1}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2^{2 (2 x + 1)} \cdot 2^{1 - x}}{8^{x + 2}}$

Apply the distributive property.

$\frac{2^{2 (2 x) + 2 \cdot 1} \cdot 2^{1 - x}}{8^{x + 2}}$

Multiply $2$ by $2$ .

$\frac{2^{4 x + 2 \cdot 1} \cdot 2^{1 - x}}{8^{x + 2}}$

Multiply $2$ by $1$ .

$\frac{2^{4 x + 2} \cdot 2^{1 - x}}{8^{x + 2}}$

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2^{4 x + 2 + 1 - x}}{8^{x + 2}}$

Step 4

Subtract $x$ from $4 x$ .

$\frac{2^{3 x + 2 + 1}}{8^{x + 2}}$

Step 5

Add $2$ and $1$ .

$\frac{2^{3 x + 3}}{8^{x + 2}}$