Solve (frac{(m^{-2}n^{-1}p^{3})^{-2}}{(mn^{2})^-3(p^-3)^4})^-2

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

Evaluate

$$\frac{1}{p^{12}m^{14}n^{16}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -3 and 4 to get -12.

$$\left(\frac{\left(m^{-2}n^{-1}p^{3}\right)^{-2}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

Expand \left(m^{-2}n^{-1}p^{3}\right)^{-2}.

$$\left(\frac{\left(m^{-2}\right)^{-2}\left(n^{-1}\right)^{-2}\left(p^{3}\right)^{-2}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

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

$$\left(\frac{m^{4}\left(n^{-1}\right)^{-2}\left(p^{3}\right)^{-2}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -2 to get 2.

$$\left(\frac{m^{4}n^{2}\left(p^{3}\right)^{-2}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\left(\frac{m^{4}n^{2}p^{-6}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

Expand \left(mn^{2}\right)^{-3}.

$$\left(\frac{m^{4}n^{2}p^{-6}}{m^{-3}\left(n^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -3 to get -6.

$$\left(\frac{m^{4}n^{2}p^{-6}}{m^{-3}n^{-6}p^{-12}}\right)^{-2}$$

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

$$\left(p^{6}m^{7}n^{8}\right)^{-2}$$

Expand \left(p^{6}m^{7}n^{8}\right)^{-2}.

$$\left(p^{6}\right)^{-2}\left(m^{7}\right)^{-2}\left(n^{8}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -2 to get -12.

$$p^{-12}\left(m^{7}\right)^{-2}\left(n^{8}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -2 to get -14.

$$p^{-12}m^{-14}\left(n^{8}\right)^{-2}$$

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

$$p^{-12}m^{-14}n^{-16}$$

Expand

$$\frac{1}{p^{12}m^{14}n^{16}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -3 and 4 to get -12.

$$\left(\frac{\left(m^{-2}n^{-1}p^{3}\right)^{-2}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

Expand \left(m^{-2}n^{-1}p^{3}\right)^{-2}.

$$\left(\frac{\left(m^{-2}\right)^{-2}\left(n^{-1}\right)^{-2}\left(p^{3}\right)^{-2}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

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

$$\left(\frac{m^{4}\left(n^{-1}\right)^{-2}\left(p^{3}\right)^{-2}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -2 to get 2.

$$\left(\frac{m^{4}n^{2}\left(p^{3}\right)^{-2}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\left(\frac{m^{4}n^{2}p^{-6}}{\left(mn^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

Expand \left(mn^{2}\right)^{-3}.

$$\left(\frac{m^{4}n^{2}p^{-6}}{m^{-3}\left(n^{2}\right)^{-3}p^{-12}}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -3 to get -6.

$$\left(\frac{m^{4}n^{2}p^{-6}}{m^{-3}n^{-6}p^{-12}}\right)^{-2}$$

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

$$\left(p^{6}m^{7}n^{8}\right)^{-2}$$

Expand \left(p^{6}m^{7}n^{8}\right)^{-2}.

$$\left(p^{6}\right)^{-2}\left(m^{7}\right)^{-2}\left(n^{8}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -2 to get -12.

$$p^{-12}\left(m^{7}\right)^{-2}\left(n^{8}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -2 to get -14.

$$p^{-12}m^{-14}\left(n^{8}\right)^{-2}$$

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

$$p^{-12}m^{-14}n^{-16}$$