Solve 5(frac{5^{3}*5^{-2}}{5^{-1}*5^{4}})^{-2}=(frac{5^{3}*5}{5^{2}*5^{4}})^-2=(5^4/5^6)^-2

$$5 ( \frac { 5 ^ { 3 } \times 5 ^ { - 2 } } { 5 ^ { - 1 } \times 5 ^ { 4 } } ) ^ { - 2 } = ( \frac { 5 ^ { 3 } \times 5 } { 5 ^ { 2 } \times 5 ^ { 4 } } ) ^ { - 2 } = ( \frac { 5 ^ { 4 } } { 5 ^ { 6 } } ) ^ { - 2 }$$

Verify

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and -2 to get 1.

$$5\times \left(\frac{5^{1}}{5^{-1}\times 5^{4}}\right)^{-2}=\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}\text{ and }\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

To multiply powers of the same base, add their exponents. Add -1 and 4 to get 3.

$$5\times \left(\frac{5^{1}}{5^{3}}\right)^{-2}=\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}\text{ and }\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

Rewrite 5^{3} as 5^{1}\times 5^{2}. Cancel out 5^{1} in both numerator and denominator.

$$5\times \left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}\text{ and }\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$5\times \left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{5^{4}}{5^{2}\times 5^{4}}\right)^{-2}\text{ and }\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$5\times \left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}\text{ and }\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

Rewrite 5^{6} as 5^{4}\times 5^{2}. Cancel out 5^{4} in both numerator and denominator.

$$5\times \left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}\text{ and }\left(\frac{5^{3}\times 5}{5^{2}\times 5^{4}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$5\times \left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}\text{ and }\left(\frac{5^{4}}{5^{2}\times 5^{4}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$5\times \left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}\text{ and }\left(\frac{5^{4}}{5^{6}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

Rewrite 5^{6} as 5^{4}\times 5^{2}. Cancel out 5^{4} in both numerator and denominator.

$$5\times \left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}\text{ and }\left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{5^{4}}{5^{6}}\right)^{-2}$$

Rewrite 5^{6} as 5^{4}\times 5^{2}. Cancel out 5^{4} in both numerator and denominator.

$$5\times \left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}\text{ and }\left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}$$

Calculate 5 to the power of 2 and get 25.

$$5\times \left(\frac{1}{25}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}\text{ and }\left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}$$

Calculate \frac{1}{25} to the power of -2 and get 625.

$$5\times 625=\left(\frac{1}{5^{2}}\right)^{-2}\text{ and }\left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}$$

Multiply 5 and 625 to get 3125.

$$3125=\left(\frac{1}{5^{2}}\right)^{-2}\text{ and }\left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}$$

Calculate 5 to the power of 2 and get 25.

$$3125=\left(\frac{1}{25}\right)^{-2}\text{ and }\left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}$$

Calculate \frac{1}{25} to the power of -2 and get 625.

$$3125=625\text{ and }\left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}$$

Compare 3125 and 625.

$$\text{false}\text{ and }\left(\frac{1}{5^{2}}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}$$

Calculate 5 to the power of 2 and get 25.

$$\text{false}\text{ and }\left(\frac{1}{25}\right)^{-2}=\left(\frac{1}{5^{2}}\right)^{-2}$$

Calculate \frac{1}{25} to the power of -2 and get 625.

$$\text{false}\text{ and }625=\left(\frac{1}{5^{2}}\right)^{-2}$$

Calculate 5 to the power of 2 and get 25.

$$\text{false}\text{ and }625=\left(\frac{1}{25}\right)^{-2}$$

Calculate \frac{1}{25} to the power of -2 and get 625.

$$\text{false}\text{ and }625=625$$

Compare 625 and 625.

$$\text{false}\text{ and }\text{true}$$

The conjunction of \text{false} and \text{true} is \text{false}.

$$\text{false}$$