Solve {x^{2/3}[x^{-1/3}y^{-1/2}(x^{2}y^-2)^-frac{2{3}}]^-frac{1{2}}}^6

$$\{ x ^ { \frac { 2 } { 3 } } [ x ^ { - \frac { 1 } { 3 } } y ^ { - \frac { 1 } { 2 } } ( x ^ { 2 } y ^ { - 2 } ) ^ { - \frac { 2 } { 3 } } ] ^ { - \frac { 1 } { 2 } } \} ^ { 6 }$$

Evaluate

$$\frac{x^{9}}{y^{\frac{5}{2}}}$$

Solution Steps

Expand \left(x^{2}y^{-2}\right)^{-\frac{2}{3}}.

$$\left(x^{\frac{2}{3}}\left(x^{-\frac{1}{3}}y^{-\frac{1}{2}}\left(x^{2}\right)^{-\frac{2}{3}}\left(y^{-2}\right)^{-\frac{2}{3}}\right)^{-\frac{1}{2}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -\frac{2}{3} to get -\frac{4}{3}.

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

To raise a power to another power, multiply the exponents. Multiply -2 and -\frac{2}{3} to get \frac{4}{3}.

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

To multiply powers of the same base, add their exponents. Add -\frac{1}{3} and -\frac{4}{3} to get -\frac{5}{3}.

$$\left(x^{\frac{2}{3}}\left(x^{-\frac{5}{3}}y^{-\frac{1}{2}}y^{\frac{4}{3}}\right)^{-\frac{1}{2}}\right)^{6}$$

To multiply powers of the same base, add their exponents. Add -\frac{1}{2} and \frac{4}{3} to get \frac{5}{6}.

$$\left(x^{\frac{2}{3}}\left(x^{-\frac{5}{3}}y^{\frac{5}{6}}\right)^{-\frac{1}{2}}\right)^{6}$$

Expand \left(x^{-\frac{5}{3}}y^{\frac{5}{6}}\right)^{-\frac{1}{2}}.

$$\left(x^{\frac{2}{3}}\left(x^{-\frac{5}{3}}\right)^{-\frac{1}{2}}\left(y^{\frac{5}{6}}\right)^{-\frac{1}{2}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{5}{3} and -\frac{1}{2} to get \frac{5}{6}.

$$\left(x^{\frac{2}{3}}x^{\frac{5}{6}}\left(y^{\frac{5}{6}}\right)^{-\frac{1}{2}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{6} and -\frac{1}{2} to get -\frac{5}{12}.

$$\left(x^{\frac{2}{3}}x^{\frac{5}{6}}y^{-\frac{5}{12}}\right)^{6}$$

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and \frac{5}{6} to get \frac{3}{2}.

$$\left(x^{\frac{3}{2}}y^{-\frac{5}{12}}\right)^{6}$$

Expand \left(x^{\frac{3}{2}}y^{-\frac{5}{12}}\right)^{6}.

$$\left(x^{\frac{3}{2}}\right)^{6}\left(y^{-\frac{5}{12}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{2} and 6 to get 9.

$$x^{9}\left(y^{-\frac{5}{12}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{5}{12} and 6 to get -\frac{5}{2}.

$$x^{9}y^{-\frac{5}{2}}$$

Expand

$$\frac{x^{9}}{y^{\frac{5}{2}}}$$

Solution Steps

Expand \left(x^{2}y^{-2}\right)^{-\frac{2}{3}}.

$$\left(x^{\frac{2}{3}}\left(x^{-\frac{1}{3}}y^{-\frac{1}{2}}\left(x^{2}\right)^{-\frac{2}{3}}\left(y^{-2}\right)^{-\frac{2}{3}}\right)^{-\frac{1}{2}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -\frac{2}{3} to get -\frac{4}{3}.

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

To raise a power to another power, multiply the exponents. Multiply -2 and -\frac{2}{3} to get \frac{4}{3}.

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

To multiply powers of the same base, add their exponents. Add -\frac{1}{3} and -\frac{4}{3} to get -\frac{5}{3}.

$$\left(x^{\frac{2}{3}}\left(x^{-\frac{5}{3}}y^{-\frac{1}{2}}y^{\frac{4}{3}}\right)^{-\frac{1}{2}}\right)^{6}$$

To multiply powers of the same base, add their exponents. Add -\frac{1}{2} and \frac{4}{3} to get \frac{5}{6}.

$$\left(x^{\frac{2}{3}}\left(x^{-\frac{5}{3}}y^{\frac{5}{6}}\right)^{-\frac{1}{2}}\right)^{6}$$

Expand \left(x^{-\frac{5}{3}}y^{\frac{5}{6}}\right)^{-\frac{1}{2}}.

$$\left(x^{\frac{2}{3}}\left(x^{-\frac{5}{3}}\right)^{-\frac{1}{2}}\left(y^{\frac{5}{6}}\right)^{-\frac{1}{2}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{5}{3} and -\frac{1}{2} to get \frac{5}{6}.

$$\left(x^{\frac{2}{3}}x^{\frac{5}{6}}\left(y^{\frac{5}{6}}\right)^{-\frac{1}{2}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{6} and -\frac{1}{2} to get -\frac{5}{12}.

$$\left(x^{\frac{2}{3}}x^{\frac{5}{6}}y^{-\frac{5}{12}}\right)^{6}$$

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and \frac{5}{6} to get \frac{3}{2}.

$$\left(x^{\frac{3}{2}}y^{-\frac{5}{12}}\right)^{6}$$

Expand \left(x^{\frac{3}{2}}y^{-\frac{5}{12}}\right)^{6}.

$$\left(x^{\frac{3}{2}}\right)^{6}\left(y^{-\frac{5}{12}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{2} and 6 to get 9.

$$x^{9}\left(y^{-\frac{5}{12}}\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{5}{12} and 6 to get -\frac{5}{2}.

$$x^{9}y^{-\frac{5}{2}}$$