Solve (p^{2}*frac{q^{-2}}{r^{-3}})^-1*((2r)^2/(pq)^-1)^2

$$( p ^ { 2 } \cdot \frac { q ^ { - 2 } } { r ^ { - 3 } } ) ^ { - 1 } \cdot ( \frac { ( 2 r ) ^ { 2 } } { ( p q ) ^ { - 1 } } ) ^ { 2 }$$

Evaluate

$$16rq^{4}$$

Solution Steps

Express p^{2}\times \frac{q^{-2}}{r^{-3}} as a single fraction.

$$\left(\frac{p^{2}q^{-2}}{r^{-3}}\right)^{-1}\times \left(\frac{\left(2r\right)^{2}}{\left(pq\right)^{-1}}\right)^{2}$$

To raise \frac{p^{2}q^{-2}}{r^{-3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \left(\frac{\left(2r\right)^{2}}{\left(pq\right)^{-1}}\right)^{2}$$

Expand \left(2r\right)^{2}.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \left(\frac{2^{2}r^{2}}{\left(pq\right)^{-1}}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \left(\frac{4r^{2}}{\left(pq\right)^{-1}}\right)^{2}$$

Expand \left(pq\right)^{-1}.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \left(\frac{4r^{2}}{p^{-1}q^{-1}}\right)^{2}$$

To raise \frac{4r^{2}}{p^{-1}q^{-1}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \frac{\left(4r^{2}\right)^{2}}{\left(p^{-1}q^{-1}\right)^{2}}$$

Multiply \frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}} times \frac{\left(4r^{2}\right)^{2}}{\left(p^{-1}q^{-1}\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}\times \left(4r^{2}\right)^{2}}{\left(r^{-3}\right)^{-1}\left(p^{-1}q^{-1}\right)^{2}}$$

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

$$\frac{\left(p^{2}q^{-2}\right)^{-1}\times \left(4r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

Expand \left(p^{2}q^{-2}\right)^{-1}.

$$\frac{\left(p^{2}\right)^{-1}\left(q^{-2}\right)^{-1}\times \left(4r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}\left(q^{-2}\right)^{-1}\times \left(4r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}q^{2}\times \left(4r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

Expand \left(4r^{2}\right)^{2}.

$$\frac{p^{-2}q^{2}\times 4^{2}\left(r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}q^{2}\times 4^{2}r^{4}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{p^{-2}q^{2}\times 16r^{4}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

Expand \left(p^{-1}q^{-1}\right)^{2}.

$$\frac{p^{-2}q^{2}\times 16r^{4}}{r^{3}\left(p^{-1}\right)^{2}\left(q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}q^{2}\times 16r^{4}}{r^{3}p^{-2}\left(q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}q^{2}\times 16r^{4}}{r^{3}p^{-2}q^{-2}}$$

Cancel out p^{-2}r^{3} in both numerator and denominator.

$$\frac{16rq^{2}}{q^{-2}}$$

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

$$16rq^{4}$$

Expand

$$16rq^{4}$$

Solution Steps

Express p^{2}\times \frac{q^{-2}}{r^{-3}} as a single fraction.

$$\left(\frac{p^{2}q^{-2}}{r^{-3}}\right)^{-1}\times \left(\frac{\left(2r\right)^{2}}{\left(pq\right)^{-1}}\right)^{2}$$

To raise \frac{p^{2}q^{-2}}{r^{-3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \left(\frac{\left(2r\right)^{2}}{\left(pq\right)^{-1}}\right)^{2}$$

Expand \left(2r\right)^{2}.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \left(\frac{2^{2}r^{2}}{\left(pq\right)^{-1}}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \left(\frac{4r^{2}}{\left(pq\right)^{-1}}\right)^{2}$$

Expand \left(pq\right)^{-1}.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \left(\frac{4r^{2}}{p^{-1}q^{-1}}\right)^{2}$$

To raise \frac{4r^{2}}{p^{-1}q^{-1}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}}\times \frac{\left(4r^{2}\right)^{2}}{\left(p^{-1}q^{-1}\right)^{2}}$$

Multiply \frac{\left(p^{2}q^{-2}\right)^{-1}}{\left(r^{-3}\right)^{-1}} times \frac{\left(4r^{2}\right)^{2}}{\left(p^{-1}q^{-1}\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(p^{2}q^{-2}\right)^{-1}\times \left(4r^{2}\right)^{2}}{\left(r^{-3}\right)^{-1}\left(p^{-1}q^{-1}\right)^{2}}$$

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

$$\frac{\left(p^{2}q^{-2}\right)^{-1}\times \left(4r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

Expand \left(p^{2}q^{-2}\right)^{-1}.

$$\frac{\left(p^{2}\right)^{-1}\left(q^{-2}\right)^{-1}\times \left(4r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}\left(q^{-2}\right)^{-1}\times \left(4r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}q^{2}\times \left(4r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

Expand \left(4r^{2}\right)^{2}.

$$\frac{p^{-2}q^{2}\times 4^{2}\left(r^{2}\right)^{2}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}q^{2}\times 4^{2}r^{4}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{p^{-2}q^{2}\times 16r^{4}}{r^{3}\left(p^{-1}q^{-1}\right)^{2}}$$

Expand \left(p^{-1}q^{-1}\right)^{2}.

$$\frac{p^{-2}q^{2}\times 16r^{4}}{r^{3}\left(p^{-1}\right)^{2}\left(q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}q^{2}\times 16r^{4}}{r^{3}p^{-2}\left(q^{-1}\right)^{2}}$$

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

$$\frac{p^{-2}q^{2}\times 16r^{4}}{r^{3}p^{-2}q^{-2}}$$

Cancel out p^{-2}r^{3} in both numerator and denominator.

$$\frac{16rq^{2}}{q^{-2}}$$

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

$$16rq^{4}$$