Solve (a^{-3}b^{2})^-2(a^-2b^4)^4

$$( a ^ { - 3 } b ^ { 2 } ) ^ { - 2 } ( a ^ { - 2 } b ^ { 4 } ) ^ { 4 }$$

Evaluate

$$\frac{b^{12}}{a^{2}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(a^{-3}b^{2}\right)^{-2}\left(a^{-2}b^{4}\right)^{4}$$

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

$$\left(a^{-3}\right)^{-2}\left(b^{2}\right)^{-2}\left(a^{-2}\right)^{4}\left(b^{4}\right)^{4}$$

Use the Commutative Property of Multiplication.

$$\left(a^{-3}\right)^{-2}\left(a^{-2}\right)^{4}\left(b^{2}\right)^{-2}\left(b^{4}\right)^{4}$$

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

$$a^{-3\left(-2\right)}a^{-2\times 4}b^{2\left(-2\right)}b^{4\times 4}$$

Multiply -3 times -2.

$$a^{6}a^{-2\times 4}b^{2\left(-2\right)}b^{4\times 4}$$

Multiply -2 times 4.

$$a^{6}a^{-8}b^{2\left(-2\right)}b^{4\times 4}$$

Multiply 2 times -2.

$$a^{6}a^{-8}b^{-4}b^{4\times 4}$$

Multiply 4 times 4.

$$a^{6}a^{-8}b^{-4}b^{16}$$

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

$$a^{6-8}b^{-4+16}$$

Add the exponents 6 and -8.

$$a^{-2}b^{-4+16}$$

Add the exponents -4 and 16.

$$a^{-2}b^{12}$$

Expand

$$\frac{b^{12}}{a^{2}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(a^{-3}b^{2}\right)^{-2}\left(a^{-2}b^{4}\right)^{4}$$

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

$$\left(a^{-3}\right)^{-2}\left(b^{2}\right)^{-2}\left(a^{-2}\right)^{4}\left(b^{4}\right)^{4}$$

Use the Commutative Property of Multiplication.

$$\left(a^{-3}\right)^{-2}\left(a^{-2}\right)^{4}\left(b^{2}\right)^{-2}\left(b^{4}\right)^{4}$$

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

$$a^{-3\left(-2\right)}a^{-2\times 4}b^{2\left(-2\right)}b^{4\times 4}$$

Multiply -3 times -2.

$$a^{6}a^{-2\times 4}b^{2\left(-2\right)}b^{4\times 4}$$

Multiply -2 times 4.

$$a^{6}a^{-8}b^{2\left(-2\right)}b^{4\times 4}$$

Multiply 2 times -2.

$$a^{6}a^{-8}b^{-4}b^{4\times 4}$$

Multiply 4 times 4.

$$a^{6}a^{-8}b^{-4}b^{16}$$

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

$$a^{6-8}b^{-4+16}$$

Add the exponents 6 and -8.

$$a^{-2}b^{-4+16}$$

Add the exponents -4 and 16.

$$a^{-2}b^{12}$$