Find the Exact Value (3^-8*7^3)^-2

${(3^{- 8} \cdot 7^{3})}^{- 2}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{3^{8}} \cdot 7^{3})}^{- 2}$

Step 2

Evaluate the exponents.

Raise $3$ to the power of $8$ .

${(\frac{1}{6561} \cdot 7^{3})}^{- 2}$

Raise $7$ to the power of $3$ .

${(\frac{1}{6561} \cdot 343)}^{- 2}$

Step 3

Combine $\frac{1}{6561}$ and $343$ .

${(\frac{343}{6561})}^{- 2}$

Step 4

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{6561}{343})}^{2}$

Step 5

Apply the product rule to $\frac{6561}{343}$ .

$\frac{6561^{2}}{343^{2}}$

Step 6

Raise $6561$ to the power of $2$ .

$\frac{43046721}{343^{2}}$

Step 7

Raise $343$ to the power of $2$ .

$\frac{43046721}{117649}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{43046721}{117649}$

Decimal Form:

$365.89109129 \dots$

Mixed Number Form:

$365 \frac{104836}{117649}$