Solve {l}{P=[8(4/5)^{-2}-(2/3)^{-3}-(8/9)^{-1}]^{(3/3sqrt{9})^-3}}{Q=[(-32)^frac{8{5}}div(32)^frac{2{5}}]^frac{1{3}}}

$$\left. \begin{array} { l } { P = [ 8 ( \frac { 4 } { 5 } ) ^ { - 2 } - ( \frac { 2 } { 3 } ) ^ { - 3 } - ( \frac { 8 } { 9 } ) ^ { - 1 } ] ^ { ( \frac { 3 } { 3 \sqrt { 9 } } ) ^ { - 3 } } } \\ { Q = [ ( - 32 ) ^ { \frac { 8 } { 5 } } \div ( 32 ) ^ { \frac { 2 } { 5 } } ] ^ { \frac { 1 } { 3 } } } \end{array} \right.$$

Solve for P, Q

$$P=2417851639229258349412352$$

$$Q=4$$

Solution Steps

Consider the first equation. Calculate \frac{4}{5} to the power of -2 and get \frac{25}{16}.

$$P=\left(8\times \frac{25}{16}-\left(\frac{2}{3}\right)^{-3}-\left(\frac{8}{9}\right)^{-1}\right)^{\left(\frac{3}{3\sqrt{9}}\right)^{-3}}$$

Multiply 8 and \frac{25}{16} to get \frac{25}{2}.

$$P=\left(\frac{25}{2}-\left(\frac{2}{3}\right)^{-3}-\left(\frac{8}{9}\right)^{-1}\right)^{\left(\frac{3}{3\sqrt{9}}\right)^{-3}}$$

Calculate \frac{2}{3} to the power of -3 and get \frac{27}{8}.

$$P=\left(\frac{25}{2}-\frac{27}{8}-\left(\frac{8}{9}\right)^{-1}\right)^{\left(\frac{3}{3\sqrt{9}}\right)^{-3}}$$

Subtract \frac{27}{8} from \frac{25}{2} to get \frac{73}{8}.

$$P=\left(\frac{73}{8}-\left(\frac{8}{9}\right)^{-1}\right)^{\left(\frac{3}{3\sqrt{9}}\right)^{-3}}$$

Calculate \frac{8}{9} to the power of -1 and get \frac{9}{8}.

$$P=\left(\frac{73}{8}-\frac{9}{8}\right)^{\left(\frac{3}{3\sqrt{9}}\right)^{-3}}$$

Subtract \frac{9}{8} from \frac{73}{8} to get 8.

$$P=8^{\left(\frac{3}{3\sqrt{9}}\right)^{-3}}$$

Calculate the square root of 9 and get 3.

$$P=8^{\left(\frac{3}{3\times 3}\right)^{-3}}$$

Multiply 3 and 3 to get 9.

$$P=8^{\left(\frac{3}{9}\right)^{-3}}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

$$P=8^{\left(\frac{1}{3}\right)^{-3}}$$

Calculate \frac{1}{3} to the power of -3 and get 27.

$$P=8^{27}$$

Calculate 8 to the power of 27 and get 2417851639229258349412352.

$$P=2417851639229258349412352$$

Consider the second equation. Calculate -32 to the power of \frac{8}{5} and get 256.

$$Q=\left(\frac{256}{32^{\frac{2}{5}}}\right)^{\frac{1}{3}}$$

Calculate 32 to the power of \frac{2}{5} and get 4.

$$Q=\left(\frac{256}{4}\right)^{\frac{1}{3}}$$

Divide 256 by 4 to get 64.

$$Q=64^{\frac{1}{3}}$$

Calculate 64 to the power of \frac{1}{3} and get 4.

$$Q=4$$

The system is now solved.

$$P=2417851639229258349412352 Q=4$$