Solve frac{(2p^{2}q^{-3}r)^-3}{(3p^-2q^3r)^2}

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

Evaluate

$$\frac{q^{3}}{72p^{2}r^{5}}$$

Solution Steps

Expand \left(2p^{2}q^{-3}r\right)^{-3}.

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

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

$$\frac{2^{-3}p^{-6}\left(q^{-3}\right)^{-3}r^{-3}}{\left(3p^{-2}q^{3}r\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -3 to get 9.

$$\frac{2^{-3}p^{-6}q^{9}r^{-3}}{\left(3p^{-2}q^{3}r\right)^{2}}$$

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{\left(3p^{-2}q^{3}r\right)^{2}}$$

Expand \left(3p^{-2}q^{3}r\right)^{2}.

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{3^{2}\left(p^{-2}\right)^{2}\left(q^{3}\right)^{2}r^{2}}$$

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

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{3^{2}p^{-4}\left(q^{3}\right)^{2}r^{2}}$$

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

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{3^{2}p^{-4}q^{6}r^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{9p^{-4}q^{6}r^{2}}$$

Cancel out q^{6} in both numerator and denominator.

$$\frac{\frac{1}{8}p^{-6}r^{-3}q^{3}}{9p^{-4}r^{2}}$$

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

$$\frac{\frac{1}{8}q^{3}}{9p^{2}r^{5}}$$

Expand

$$\frac{q^{3}}{72p^{2}r^{5}}$$

Solution Steps

Expand \left(2p^{2}q^{-3}r\right)^{-3}.

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

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

$$\frac{2^{-3}p^{-6}\left(q^{-3}\right)^{-3}r^{-3}}{\left(3p^{-2}q^{3}r\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -3 to get 9.

$$\frac{2^{-3}p^{-6}q^{9}r^{-3}}{\left(3p^{-2}q^{3}r\right)^{2}}$$

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{\left(3p^{-2}q^{3}r\right)^{2}}$$

Expand \left(3p^{-2}q^{3}r\right)^{2}.

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{3^{2}\left(p^{-2}\right)^{2}\left(q^{3}\right)^{2}r^{2}}$$

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

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{3^{2}p^{-4}\left(q^{3}\right)^{2}r^{2}}$$

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

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{3^{2}p^{-4}q^{6}r^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{1}{8}p^{-6}q^{9}r^{-3}}{9p^{-4}q^{6}r^{2}}$$

Cancel out q^{6} in both numerator and denominator.

$$\frac{\frac{1}{8}p^{-6}r^{-3}q^{3}}{9p^{-4}r^{2}}$$

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

$$\frac{\frac{1}{8}q^{3}}{9p^{2}r^{5}}$$