Evaluate ((2^5)^3*2^-7)/(2^-6)

$\frac{{(2^{5})}^{3} \cdot 2^{- 7}}{2^{- 6}}$

Step 1

Move $2^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(2^{5})}^{3}}{2^{- 6} \cdot 2^{7}}$

Step 2

Simplify the denominator.

Multiply $2^{- 6}$ by $2^{7}$ by adding the exponents.

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

$\frac{{(2^{5})}^{3}}{2^{- 6 + 7}}$

Add $- 6$ and $7$ .

$\frac{{(2^{5})}^{3}}{2^{1}}$

Simplify $2^{1}$ .

$\frac{{(2^{5})}^{3}}{2}$

Step 3

Simplify the numerator.

Multiply the exponents in ${(2^{5})}^{3}$ .

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

$\frac{2^{5 \cdot 3}}{2}$

Multiply $5$ by $3$ .

$\frac{2^{15}}{2}$

Raise $2$ to the power of $15$ .

$\frac{32768}{2}$

Step 4

Divide $32768$ by $2$ .

$16384$