Solve frac{(2p^{-3}q^{6})^{3}(q^{-5}r^{6})^{-2}}{(p^{2}q^{-3})^{-4}(-3p^{-3}q^5r^-2)^2})^2

$$\frac { ( 2 p ^ { - 3 } q ^ { 6 } ) ^ { 3 } ( q ^ { - 5 } r ^ { 6 } ) ^ { - 2 } } { ( p ^ { 2 } q ^ { - 3 } ) ^ { - 4 } ( - 3 p ^ { - 3 } q ^ { 5 } r ^ { - 2 } ) ^ { 2 } } ) ^ { 2 }$$

Evaluate

$$\frac{8p^{5}q^{6}}{9r^{8}}$$

Solution Steps

Expand \left(2p^{-3}q^{6}\right)^{3}.

$$\frac{2^{3}\left(p^{-3}\right)^{3}\left(q^{6}\right)^{3}\left(q^{-5}r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

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

$$\frac{2^{3}p^{-9}\left(q^{6}\right)^{3}\left(q^{-5}r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$\frac{2^{3}p^{-9}q^{18}\left(q^{-5}r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{8p^{-9}q^{18}\left(q^{-5}r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

Expand \left(q^{-5}r^{6}\right)^{-2}.

$$\frac{8p^{-9}q^{18}\left(q^{-5}\right)^{-2}\left(r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

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

$$\frac{8p^{-9}q^{18}q^{10}\left(r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -2 to get -12.

$$\frac{8p^{-9}q^{18}q^{10}r^{-12}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 18 and 10 to get 28.

$$\frac{8p^{-9}q^{28}r^{-12}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

Expand \left(p^{2}q^{-3}\right)^{-4}.

$$\frac{8p^{-9}q^{28}r^{-12}}{\left(p^{2}\right)^{-4}\left(q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -4 to get -8.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}\left(q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -4 to get 12.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

Expand \left(-3p^{-3}q^{5}r^{-2}\right)^{2}.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3\right)^{2}\left(p^{-3}\right)^{2}\left(q^{5}\right)^{2}\left(r^{-2}\right)^{2}}$$

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

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3\right)^{2}p^{-6}\left(q^{5}\right)^{2}\left(r^{-2}\right)^{2}}$$

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

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3\right)^{2}p^{-6}q^{10}\left(r^{-2}\right)^{2}}$$

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

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3\right)^{2}p^{-6}q^{10}r^{-4}}$$

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

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\times 9p^{-6}q^{10}r^{-4}}$$

To multiply powers of the same base, add their exponents. Add -8 and -6 to get -14.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-14}q^{12}\times 9q^{10}r^{-4}}$$

To multiply powers of the same base, add their exponents. Add 12 and 10 to get 22.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-14}q^{22}\times 9r^{-4}}$$

Cancel out q^{22} in both numerator and denominator.

$$\frac{8r^{-12}p^{-9}q^{6}}{9p^{-14}r^{-4}}$$

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

$$\frac{8r^{-12}p^{5}q^{6}}{9r^{-4}}$$

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

$$\frac{8p^{5}q^{6}}{9r^{8}}$$

Expand

$$\frac{8p^{5}q^{6}}{9r^{8}}$$

Solution Steps

Expand \left(2p^{-3}q^{6}\right)^{3}.

$$\frac{2^{3}\left(p^{-3}\right)^{3}\left(q^{6}\right)^{3}\left(q^{-5}r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

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

$$\frac{2^{3}p^{-9}\left(q^{6}\right)^{3}\left(q^{-5}r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$\frac{2^{3}p^{-9}q^{18}\left(q^{-5}r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{8p^{-9}q^{18}\left(q^{-5}r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

Expand \left(q^{-5}r^{6}\right)^{-2}.

$$\frac{8p^{-9}q^{18}\left(q^{-5}\right)^{-2}\left(r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

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

$$\frac{8p^{-9}q^{18}q^{10}\left(r^{6}\right)^{-2}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -2 to get -12.

$$\frac{8p^{-9}q^{18}q^{10}r^{-12}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 18 and 10 to get 28.

$$\frac{8p^{-9}q^{28}r^{-12}}{\left(p^{2}q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

Expand \left(p^{2}q^{-3}\right)^{-4}.

$$\frac{8p^{-9}q^{28}r^{-12}}{\left(p^{2}\right)^{-4}\left(q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -4 to get -8.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}\left(q^{-3}\right)^{-4}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -4 to get 12.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3p^{-3}q^{5}r^{-2}\right)^{2}}$$

Expand \left(-3p^{-3}q^{5}r^{-2}\right)^{2}.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3\right)^{2}\left(p^{-3}\right)^{2}\left(q^{5}\right)^{2}\left(r^{-2}\right)^{2}}$$

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

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3\right)^{2}p^{-6}\left(q^{5}\right)^{2}\left(r^{-2}\right)^{2}}$$

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

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3\right)^{2}p^{-6}q^{10}\left(r^{-2}\right)^{2}}$$

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

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\left(-3\right)^{2}p^{-6}q^{10}r^{-4}}$$

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

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-8}q^{12}\times 9p^{-6}q^{10}r^{-4}}$$

To multiply powers of the same base, add their exponents. Add -8 and -6 to get -14.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-14}q^{12}\times 9q^{10}r^{-4}}$$

To multiply powers of the same base, add their exponents. Add 12 and 10 to get 22.

$$\frac{8p^{-9}q^{28}r^{-12}}{p^{-14}q^{22}\times 9r^{-4}}$$

Cancel out q^{22} in both numerator and denominator.

$$\frac{8r^{-12}p^{-9}q^{6}}{9p^{-14}r^{-4}}$$

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

$$\frac{8r^{-12}p^{5}q^{6}}{9r^{-4}}$$

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

$$\frac{8p^{5}q^{6}}{9r^{8}}$$