Solve (m^{-4}n)^3(m^6n^2)^3

$$( m ^ { - 4 } n ) ^ { 3 } ( m ^ { 6 } n ^ { 2 } ) ^ { 3 }$$

Evaluate

$$m^{6}n^{9}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(m^{-4}n^{1}\right)^{3}\left(m^{6}n^{2}\right)^{3}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\left(m^{-4}\right)^{3}\left(n^{1}\right)^{3}\left(m^{6}\right)^{3}\left(n^{2}\right)^{3}$$

Use the Commutative Property of Multiplication.

$$\left(m^{-4}\right)^{3}\left(m^{6}\right)^{3}\left(n^{1}\right)^{3}\left(n^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents.

$$m^{-4\times 3}m^{6\times 3}n^{3}n^{2\times 3}$$

Multiply -4 times 3.

$$m^{-12}m^{6\times 3}n^{3}n^{2\times 3}$$

Multiply 6 times 3.

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

Multiply 2 times 3.

$$m^{-12}m^{18}n^{3}n^{6}$$

To multiply powers of the same base, add their exponents.

$$m^{-12+18}n^{3+6}$$

Add the exponents -12 and 18.

$$m^{6}n^{3+6}$$

Add the exponents 3 and 6.

$$m^{6}n^{9}$$

Expand

$$m^{6}n^{9}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(m^{-4}n^{1}\right)^{3}\left(m^{6}n^{2}\right)^{3}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\left(m^{-4}\right)^{3}\left(n^{1}\right)^{3}\left(m^{6}\right)^{3}\left(n^{2}\right)^{3}$$

Use the Commutative Property of Multiplication.

$$\left(m^{-4}\right)^{3}\left(m^{6}\right)^{3}\left(n^{1}\right)^{3}\left(n^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents.

$$m^{-4\times 3}m^{6\times 3}n^{3}n^{2\times 3}$$

Multiply -4 times 3.

$$m^{-12}m^{6\times 3}n^{3}n^{2\times 3}$$

Multiply 6 times 3.

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

Multiply 2 times 3.

$$m^{-12}m^{18}n^{3}n^{6}$$

To multiply powers of the same base, add their exponents.

$$m^{-12+18}n^{3+6}$$

Add the exponents -12 and 18.

$$m^{6}n^{3+6}$$

Add the exponents 3 and 6.

$$m^{6}n^{9}$$