Find the Exact Value ((2^-10)/(4^2))^7

${(\frac{2^{- 10}}{4^{2}})}^{7}$

Step 1

Move $2^{- 10}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{4^{2} \cdot 2^{10}})}^{7}$

Step 2

Simplify the denominator.

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

${(\frac{1}{{(2^{2})}^{2} \cdot 2^{10}})}^{7}$

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

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

${(\frac{1}{2^{2 \cdot 2} \cdot 2^{10}})}^{7}$

Multiply $2$ by $2$ .

${(\frac{1}{2^{4} \cdot 2^{10}})}^{7}$

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

${(\frac{1}{2^{4 + 10}})}^{7}$

Add $4$ and $10$ .

${(\frac{1}{2^{14}})}^{7}$

Step 3

Raise $2$ to the power of $14$ .

${(\frac{1}{16384})}^{7}$

Step 4

Apply the product rule to $\frac{1}{16384}$ .

$\frac{1^{7}}{16384^{7}}$

Step 5

One to any power is one.

$\frac{1}{16384^{7}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{16384^{7}}$

Decimal Form:

$3.15544362 \cdot 10^{- 30}$