Simplify (3^-4+5^-3)^-2

${(3^{- 4} + 5^{- 3})}^{- 2}$

Step 1

Simplify each term.

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

${(\frac{1}{3^{4}} + 5^{- 3})}^{- 2}$

Raise $3$ to the power of $4$ .

${(\frac{1}{81} + 5^{- 3})}^{- 2}$

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

${(\frac{1}{81} + \frac{1}{5^{3}})}^{- 2}$

Raise $5$ to the power of $3$ .

${(\frac{1}{81} + \frac{1}{125})}^{- 2}$

Step 2

To write $\frac{1}{81}$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

${(\frac{1}{81} \cdot \frac{125}{125} + \frac{1}{125})}^{- 2}$

Step 3

To write $\frac{1}{125}$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

${(\frac{1}{81} \cdot \frac{125}{125} + \frac{1}{125} \cdot \frac{81}{81})}^{- 2}$

Step 4

Write each expression with a common denominator of $10125$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{81}$ by $\frac{125}{125}$ .

${(\frac{125}{81 \cdot 125} + \frac{1}{125} \cdot \frac{81}{81})}^{- 2}$

Multiply $81$ by $125$ .

${(\frac{125}{10125} + \frac{1}{125} \cdot \frac{81}{81})}^{- 2}$

Multiply $\frac{1}{125}$ by $\frac{81}{81}$ .

${(\frac{125}{10125} + \frac{81}{125 \cdot 81})}^{- 2}$

Multiply $125$ by $81$ .

${(\frac{125}{10125} + \frac{81}{10125})}^{- 2}$

Step 5

Combine the numerators over the common denominator.

${(\frac{125 + 81}{10125})}^{- 2}$

Step 6

Add $125$ and $81$ .

${(\frac{206}{10125})}^{- 2}$

Step 7

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

${(\frac{10125}{206})}^{2}$

Step 8

Apply the product rule to $\frac{10125}{206}$ .

$\frac{10125^{2}}{206^{2}}$

Step 9

Raise $10125$ to the power of $2$ .

$\frac{102515625}{206^{2}}$

Step 10

Raise $206$ to the power of $2$ .

$\frac{102515625}{42436}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{102515625}{42436}$

Decimal Form:

$2415.77021868 \dots$

Mixed Number Form:

$2415 \frac{32685}{42436}$

Simplify (3^-4+5^-3)^-2

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