Solve (a^{3}b)^-1*(2a^-5b^-2)^-2

$$( a ^ { 3 } b ) ^ { - 1 } \cdot ( 2 a ^ { - 5 } b ^ { - 2 } ) ^ { - 2 }$$

Evaluate

$$\frac{b^{3}a^{7}}{4}$$

Solution Steps

Expand \left(a^{3}b\right)^{-1}.

$$\left(a^{3}\right)^{-1}b^{-1}\times \left(2a^{-5}b^{-2}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$a^{-3}b^{-1}\times \left(2a^{-5}b^{-2}\right)^{-2}$$

Expand \left(2a^{-5}b^{-2}\right)^{-2}.

$$a^{-3}b^{-1}\times 2^{-2}\left(a^{-5}\right)^{-2}\left(b^{-2}\right)^{-2}$$

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

$$a^{-3}b^{-1}\times 2^{-2}a^{10}\left(b^{-2}\right)^{-2}$$

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

$$a^{-3}b^{-1}\times 2^{-2}a^{10}b^{4}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$a^{-3}b^{-1}\times \frac{1}{4}a^{10}b^{4}$$

To multiply powers of the same base, add their exponents. Add -3 and 10 to get 7.

$$a^{7}b^{-1}\times \frac{1}{4}b^{4}$$

To multiply powers of the same base, add their exponents. Add -1 and 4 to get 3.

$$a^{7}b^{3}\times \frac{1}{4}$$

Expand

$$\frac{b^{3}a^{7}}{4}$$

Solution Steps

Expand \left(a^{3}b\right)^{-1}.

$$\left(a^{3}\right)^{-1}b^{-1}\times \left(2a^{-5}b^{-2}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$a^{-3}b^{-1}\times \left(2a^{-5}b^{-2}\right)^{-2}$$

Expand \left(2a^{-5}b^{-2}\right)^{-2}.

$$a^{-3}b^{-1}\times 2^{-2}\left(a^{-5}\right)^{-2}\left(b^{-2}\right)^{-2}$$

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

$$a^{-3}b^{-1}\times 2^{-2}a^{10}\left(b^{-2}\right)^{-2}$$

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

$$a^{-3}b^{-1}\times 2^{-2}a^{10}b^{4}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$a^{-3}b^{-1}\times \frac{1}{4}a^{10}b^{4}$$

To multiply powers of the same base, add their exponents. Add -3 and 10 to get 7.

$$a^{7}b^{-1}\times \frac{1}{4}b^{4}$$

To multiply powers of the same base, add their exponents. Add -1 and 4 to get 3.

$$a^{7}b^{3}\times \frac{1}{4}$$