Solve (a^{3}b^{-1})^-2(a^-2b^2)^2

$$( a ^ { 3 } b ^ { - 1 } ) ^ { - 2 } ( a ^ { - 2 } b ^ { 2 } ) ^ { 2 }$$

Evaluate

$$\frac{b^{6}}{a^{10}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(a^{3}\times \frac{1}{b}\right)^{-2}\left(a^{-2}b^{2}\right)^{2}$$

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

$$\left(a^{3}\right)^{-2}\times \left(\frac{1}{b}\right)^{-2}\left(a^{-2}\right)^{2}\left(b^{2}\right)^{2}$$

Use the Commutative Property of Multiplication.

$$\left(a^{3}\right)^{-2}\left(a^{-2}\right)^{2}\times \left(\frac{1}{b}\right)^{-2}\left(b^{2}\right)^{2}$$

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

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

Multiply 3 times -2.

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

Multiply -2 times 2.

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

Multiply -1 times -2.

$$a^{-6}a^{-4}b^{2}b^{2\times 2}$$

Multiply 2 times 2.

$$a^{-6}a^{-4}b^{2}b^{4}$$

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

$$a^{-6-4}b^{2+4}$$

Add the exponents -6 and -4.

$$a^{-10}b^{2+4}$$

Add the exponents 2 and 4.

$$a^{-10}b^{6}$$

Expand

$$\frac{b^{6}}{a^{10}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(a^{3}\times \frac{1}{b}\right)^{-2}\left(a^{-2}b^{2}\right)^{2}$$

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

$$\left(a^{3}\right)^{-2}\times \left(\frac{1}{b}\right)^{-2}\left(a^{-2}\right)^{2}\left(b^{2}\right)^{2}$$

Use the Commutative Property of Multiplication.

$$\left(a^{3}\right)^{-2}\left(a^{-2}\right)^{2}\times \left(\frac{1}{b}\right)^{-2}\left(b^{2}\right)^{2}$$

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

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

Multiply 3 times -2.

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

Multiply -2 times 2.

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

Multiply -1 times -2.

$$a^{-6}a^{-4}b^{2}b^{2\times 2}$$

Multiply 2 times 2.

$$a^{-6}a^{-4}b^{2}b^{4}$$

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

$$a^{-6-4}b^{2+4}$$

Add the exponents -6 and -4.

$$a^{-10}b^{2+4}$$

Add the exponents 2 and 4.

$$a^{-10}b^{6}$$