Solve left(2g^-2h^-1right)^2div4g^2h/{left(-3{h^3{g}^-1right)}^2}

$${ \left(2 { g }^{ -2 } { h }^{ -1 } \right) }^{ 2 } \div \frac{ 4 { g }^{ 2 } h }{ { \left(-3 { h }^{ 3 } { g }^{ -1 } \right) }^{ 2 } }$$

Evaluate

$$\frac{9h^{3}}{g^{8}}$$

Solution Steps

Expand \left(2g^{-2}h^{-1}\right)^{2}.

$$\frac{2^{2}\left(g^{-2}\right)^{2}\left(h^{-1}\right)^{2}}{\frac{4g^{2}h}{\left(-3h^{3}g^{-1}\right)^{2}}}$$

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

$$\frac{2^{2}g^{-4}\left(h^{-1}\right)^{2}}{\frac{4g^{2}h}{\left(-3h^{3}g^{-1}\right)^{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{2^{2}g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3h^{3}g^{-1}\right)^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3h^{3}g^{-1}\right)^{2}}}$$

Expand \left(-3h^{3}g^{-1}\right)^{2}.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3\right)^{2}\left(h^{3}\right)^{2}\left(g^{-1}\right)^{2}}}$$

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

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3\right)^{2}h^{6}\left(g^{-1}\right)^{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3\right)^{2}h^{6}g^{-2}}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{9h^{6}g^{-2}}}$$

Cancel out h in both numerator and denominator.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}}{9g^{-2}h^{5}}}$$

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

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{4}}{9h^{5}}}$$

Divide 4g^{-4}h^{-2} by \frac{4g^{4}}{9h^{5}} by multiplying 4g^{-4}h^{-2} by the reciprocal of \frac{4g^{4}}{9h^{5}}.

$$\frac{4g^{-4}h^{-2}\times 9h^{5}}{4g^{4}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{9g^{-4}h^{-2}h^{5}}{g^{4}}$$

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

$$\frac{9h^{-2}h^{5}}{g^{8}}$$

To multiply powers of the same base, add their exponents. Add -2 and 5 to get 3.

$$\frac{9h^{3}}{g^{8}}$$

Expand

$$\frac{9h^{3}}{g^{8}}$$

Solution Steps

Expand \left(2g^{-2}h^{-1}\right)^{2}.

$$\frac{2^{2}\left(g^{-2}\right)^{2}\left(h^{-1}\right)^{2}}{\frac{4g^{2}h}{\left(-3h^{3}g^{-1}\right)^{2}}}$$

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

$$\frac{2^{2}g^{-4}\left(h^{-1}\right)^{2}}{\frac{4g^{2}h}{\left(-3h^{3}g^{-1}\right)^{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{2^{2}g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3h^{3}g^{-1}\right)^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3h^{3}g^{-1}\right)^{2}}}$$

Expand \left(-3h^{3}g^{-1}\right)^{2}.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3\right)^{2}\left(h^{3}\right)^{2}\left(g^{-1}\right)^{2}}}$$

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

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3\right)^{2}h^{6}\left(g^{-1}\right)^{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{\left(-3\right)^{2}h^{6}g^{-2}}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}h}{9h^{6}g^{-2}}}$$

Cancel out h in both numerator and denominator.

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{2}}{9g^{-2}h^{5}}}$$

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

$$\frac{4g^{-4}h^{-2}}{\frac{4g^{4}}{9h^{5}}}$$

Divide 4g^{-4}h^{-2} by \frac{4g^{4}}{9h^{5}} by multiplying 4g^{-4}h^{-2} by the reciprocal of \frac{4g^{4}}{9h^{5}}.

$$\frac{4g^{-4}h^{-2}\times 9h^{5}}{4g^{4}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{9g^{-4}h^{-2}h^{5}}{g^{4}}$$

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

$$\frac{9h^{-2}h^{5}}{g^{8}}$$

To multiply powers of the same base, add their exponents. Add -2 and 5 to get 3.

$$\frac{9h^{3}}{g^{8}}$$