Solve left(ab^2right)^3left(-aright)^-21/{left(b^-2right)^5}

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

Evaluate

$$ab^{16}$$

Solution Steps

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

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

Expand \left(ab^{2}\right)^{3}.

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

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

$$a^{3}b^{6}\left(-a\right)^{-2}\times \frac{1}{b^{-10}}$$

Express a^{3}\times \frac{1}{b^{-10}} as a single fraction.

$$\frac{a^{3}}{b^{-10}}b^{6}\left(-a\right)^{-2}$$

Express \frac{a^{3}}{b^{-10}}b^{6} as a single fraction.

$$\frac{a^{3}b^{6}}{b^{-10}}\left(-a\right)^{-2}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$a^{3}b^{16}\left(-a\right)^{-2}$$

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

$$a^{3}b^{16}\left(-1\right)^{-2}a^{-2}$$

Calculate -1 to the power of -2 and get 1.

$$a^{3}b^{16}\times 1a^{-2}$$

To multiply powers of the same base, add their exponents. Add 3 and -2 to get 1.

$$a^{1}b^{16}\times 1$$

Calculate a to the power of 1 and get a.

$$ab^{16}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$ab^{16}$$

Expand

$$ab^{16}$$

Solution Steps

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

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

Expand \left(ab^{2}\right)^{3}.

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

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

$$a^{3}b^{6}\left(-a\right)^{-2}\times \frac{1}{b^{-10}}$$

Express a^{3}\times \frac{1}{b^{-10}} as a single fraction.

$$\frac{a^{3}}{b^{-10}}b^{6}\left(-a\right)^{-2}$$

Express \frac{a^{3}}{b^{-10}}b^{6} as a single fraction.

$$\frac{a^{3}b^{6}}{b^{-10}}\left(-a\right)^{-2}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$a^{3}b^{16}\left(-a\right)^{-2}$$

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

$$a^{3}b^{16}\left(-1\right)^{-2}a^{-2}$$

Calculate -1 to the power of -2 and get 1.

$$a^{3}b^{16}\times 1a^{-2}$$

To multiply powers of the same base, add their exponents. Add 3 and -2 to get 1.

$$a^{1}b^{16}\times 1$$

Calculate a to the power of 1 and get a.

$$ab^{16}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$ab^{16}$$