Solve (12m^{-3}n^{5})^-2*(18m^5n^-3)^3

$$( 12 m ^ { - 3 } n ^ { 5 } ) ^ { - 2 } \times ( 18 m ^ { 5 } n ^ { - 3 } ) ^ { 3 }$$

Evaluate

$$\frac{81m^{21}}{2n^{19}}$$

Solution Steps

Expand \left(12m^{-3}n^{5}\right)^{-2}.

$$12^{-2}\left(m^{-3}\right)^{-2}\left(n^{5}\right)^{-2}\times \left(18m^{5}n^{-3}\right)^{3}$$

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

$$12^{-2}m^{6}\left(n^{5}\right)^{-2}\times \left(18m^{5}n^{-3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -2 to get -10.

$$12^{-2}m^{6}n^{-10}\times \left(18m^{5}n^{-3}\right)^{3}$$

Calculate 12 to the power of -2 and get \frac{1}{144}.

$$\frac{1}{144}m^{6}n^{-10}\times \left(18m^{5}n^{-3}\right)^{3}$$

Expand \left(18m^{5}n^{-3}\right)^{3}.

$$\frac{1}{144}m^{6}n^{-10}\times 18^{3}\left(m^{5}\right)^{3}\left(n^{-3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{1}{144}m^{6}n^{-10}\times 18^{3}m^{15}\left(n^{-3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{1}{144}m^{6}n^{-10}\times 18^{3}m^{15}n^{-9}$$

Calculate 18 to the power of 3 and get 5832.

$$\frac{1}{144}m^{6}n^{-10}\times 5832m^{15}n^{-9}$$

Multiply \frac{1}{144} and 5832 to get \frac{81}{2}.

$$\frac{81}{2}m^{6}n^{-10}m^{15}n^{-9}$$

To multiply powers of the same base, add their exponents. Add 6 and 15 to get 21.

$$\frac{81}{2}m^{21}n^{-10}n^{-9}$$

To multiply powers of the same base, add their exponents. Add -10 and -9 to get -19.

$$\frac{81}{2}m^{21}n^{-19}$$

Expand

$$\frac{81m^{21}}{2n^{19}}$$

Solution Steps

Expand \left(12m^{-3}n^{5}\right)^{-2}.

$$12^{-2}\left(m^{-3}\right)^{-2}\left(n^{5}\right)^{-2}\times \left(18m^{5}n^{-3}\right)^{3}$$

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

$$12^{-2}m^{6}\left(n^{5}\right)^{-2}\times \left(18m^{5}n^{-3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -2 to get -10.

$$12^{-2}m^{6}n^{-10}\times \left(18m^{5}n^{-3}\right)^{3}$$

Calculate 12 to the power of -2 and get \frac{1}{144}.

$$\frac{1}{144}m^{6}n^{-10}\times \left(18m^{5}n^{-3}\right)^{3}$$

Expand \left(18m^{5}n^{-3}\right)^{3}.

$$\frac{1}{144}m^{6}n^{-10}\times 18^{3}\left(m^{5}\right)^{3}\left(n^{-3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{1}{144}m^{6}n^{-10}\times 18^{3}m^{15}\left(n^{-3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{1}{144}m^{6}n^{-10}\times 18^{3}m^{15}n^{-9}$$

Calculate 18 to the power of 3 and get 5832.

$$\frac{1}{144}m^{6}n^{-10}\times 5832m^{15}n^{-9}$$

Multiply \frac{1}{144} and 5832 to get \frac{81}{2}.

$$\frac{81}{2}m^{6}n^{-10}m^{15}n^{-9}$$

To multiply powers of the same base, add their exponents. Add 6 and 15 to get 21.

$$\frac{81}{2}m^{21}n^{-10}n^{-9}$$

To multiply powers of the same base, add their exponents. Add -10 and -9 to get -19.

$$\frac{81}{2}m^{21}n^{-19}$$