Solve frac{(3m^{-8}n^{4})^{-2}}{(3^2m^-1n^5)^-8}

$$\frac { ( 3 m ^ { - 8 } n ^ { 4 } ) ^ { - 2 } } { ( 3 ^ { 2 } m ^ { - 1 } n ^ { 5 } ) ^ { - 8 } }$$

Evaluate

$$4782969m^{8}n^{32}$$

Solution Steps

Expand \left(3m^{-8}n^{4}\right)^{-2}.

$$\frac{3^{-2}\left(m^{-8}\right)^{-2}\left(n^{4}\right)^{-2}}{\left(3^{2}m^{-1}n^{5}\right)^{-8}}$$

To raise a power to another power, multiply the exponents. Multiply -8 and -2 to get 16.

$$\frac{3^{-2}m^{16}\left(n^{4}\right)^{-2}}{\left(3^{2}m^{-1}n^{5}\right)^{-8}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -2 to get -8.

$$\frac{3^{-2}m^{16}n^{-8}}{\left(3^{2}m^{-1}n^{5}\right)^{-8}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{\left(3^{2}m^{-1}n^{5}\right)^{-8}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{\left(9m^{-1}n^{5}\right)^{-8}}$$

Expand \left(9m^{-1}n^{5}\right)^{-8}.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{9^{-8}\left(m^{-1}\right)^{-8}\left(n^{5}\right)^{-8}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -8 to get 8.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{9^{-8}m^{8}\left(n^{5}\right)^{-8}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -8 to get -40.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{9^{-8}m^{8}n^{-40}}$$

Calculate 9 to the power of -8 and get \frac{1}{43046721}.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{\frac{1}{43046721}m^{8}n^{-40}}$$

Cancel out m^{8} in both numerator and denominator.

$$\frac{\frac{1}{9}n^{-8}m^{8}}{\frac{1}{43046721}n^{-40}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\frac{1}{9}m^{8}n^{32}}{\frac{1}{43046721}}$$

Divide \frac{1}{9}m^{8}n^{32} by \frac{1}{43046721} by multiplying \frac{1}{9}m^{8}n^{32} by the reciprocal of \frac{1}{43046721}.

$$\frac{1}{9}m^{8}n^{32}\times 43046721$$

Multiply \frac{1}{9} and 43046721 to get 4782969.

$$4782969m^{8}n^{32}$$

Expand

$$4782969m^{8}n^{32}$$

Solution Steps

Expand \left(3m^{-8}n^{4}\right)^{-2}.

$$\frac{3^{-2}\left(m^{-8}\right)^{-2}\left(n^{4}\right)^{-2}}{\left(3^{2}m^{-1}n^{5}\right)^{-8}}$$

To raise a power to another power, multiply the exponents. Multiply -8 and -2 to get 16.

$$\frac{3^{-2}m^{16}\left(n^{4}\right)^{-2}}{\left(3^{2}m^{-1}n^{5}\right)^{-8}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -2 to get -8.

$$\frac{3^{-2}m^{16}n^{-8}}{\left(3^{2}m^{-1}n^{5}\right)^{-8}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{\left(3^{2}m^{-1}n^{5}\right)^{-8}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{\left(9m^{-1}n^{5}\right)^{-8}}$$

Expand \left(9m^{-1}n^{5}\right)^{-8}.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{9^{-8}\left(m^{-1}\right)^{-8}\left(n^{5}\right)^{-8}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -8 to get 8.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{9^{-8}m^{8}\left(n^{5}\right)^{-8}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -8 to get -40.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{9^{-8}m^{8}n^{-40}}$$

Calculate 9 to the power of -8 and get \frac{1}{43046721}.

$$\frac{\frac{1}{9}m^{16}n^{-8}}{\frac{1}{43046721}m^{8}n^{-40}}$$

Cancel out m^{8} in both numerator and denominator.

$$\frac{\frac{1}{9}n^{-8}m^{8}}{\frac{1}{43046721}n^{-40}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\frac{1}{9}m^{8}n^{32}}{\frac{1}{43046721}}$$

Divide \frac{1}{9}m^{8}n^{32} by \frac{1}{43046721} by multiplying \frac{1}{9}m^{8}n^{32} by the reciprocal of \frac{1}{43046721}.

$$\frac{1}{9}m^{8}n^{32}\times 43046721$$

Multiply \frac{1}{9} and 43046721 to get 4782969.

$$4782969m^{8}n^{32}$$