Solve {left(frac{{left(w^-1x^3z^2right)}^{3/5}{left(w^3{x}^{3}{y}^{1/3}right)}^{1/5}}{{left({w}^{-4}{x}^{-1}{y}^2{z}^-frac{1{5}}right)}^-4}right)}^54

$${ \left( \frac{ { \left( { w }^{ -1 } { x }^{ 3 } { z }^{ 2 } \right) }^{ \frac{ 3 }{ 5 } } { \left( { w }^{ 3 } { x }^{ 3 } { y }^{ \frac{ 1 }{ 3 } } \right) }^{ \frac{ 1 }{ 5 } } }{ { \left( { w }^{ -4 } { x }^{ -1 } { y }^{ 2 } { z }^{ - \frac{ 1 }{ 5 } } \right) }^{ -4 } } \right) }^{ 54 }$$

Evaluate

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{x^{\frac{432}{5}}w^{864}}$$

Solution Steps

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

$$\left(\frac{\left(w^{-1}\right)^{\frac{3}{5}}\left(x^{3}\right)^{\frac{3}{5}}\left(z^{2}\right)^{\frac{3}{5}}\left(w^{3}x^{3}y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}\left(x^{3}\right)^{\frac{3}{5}}\left(z^{2}\right)^{\frac{3}{5}}\left(w^{3}x^{3}y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply 3 and \frac{3}{5} to get \frac{9}{5}.

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}\left(z^{2}\right)^{\frac{3}{5}}\left(w^{3}x^{3}y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}\left(w^{3}x^{3}y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}\left(w^{3}\right)^{\frac{1}{5}}\left(x^{3}\right)^{\frac{1}{5}}\left(y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}w^{\frac{3}{5}}\left(x^{3}\right)^{\frac{1}{5}}\left(y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}w^{\frac{3}{5}}x^{\frac{3}{5}}\left(y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and \frac{1}{5} to get \frac{1}{15}.

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}w^{\frac{3}{5}}x^{\frac{3}{5}}y^{\frac{1}{15}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

Multiply w^{-\frac{3}{5}} and w^{\frac{3}{5}} to get 1.

$$\left(\frac{x^{\frac{9}{5}}z^{\frac{6}{5}}x^{\frac{3}{5}}y^{\frac{1}{15}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To multiply powers of the same base, add their exponents. Add \frac{9}{5} and \frac{3}{5} to get \frac{12}{5}.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

Expand \left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{\left(w^{-4}\right)^{-4}\left(x^{-1}\right)^{-4}\left(y^{2}\right)^{-4}\left(z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply -4 and -4 to get 16.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{w^{16}\left(x^{-1}\right)^{-4}\left(y^{2}\right)^{-4}\left(z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -4 to get 4.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{w^{16}x^{4}\left(y^{2}\right)^{-4}\left(z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{w^{16}x^{4}y^{-8}\left(z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{1}{5} and -4 to get \frac{4}{5}.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{w^{16}x^{4}y^{-8}z^{\frac{4}{5}}}\right)^{54}$$

Cancel out z^{\frac{4}{5}}x^{\frac{12}{5}} in both numerator and denominator.

$$\left(\frac{\sqrt[15]{y}z^{\frac{2}{5}}}{y^{-8}x^{\frac{8}{5}}w^{16}}\right)^{54}$$

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

$$\left(\frac{z^{\frac{2}{5}}y^{\frac{121}{15}}}{x^{\frac{8}{5}}w^{16}}\right)^{54}$$

To raise \frac{z^{\frac{2}{5}}y^{\frac{121}{15}}}{x^{\frac{8}{5}}w^{16}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(z^{\frac{2}{5}}y^{\frac{121}{15}}\right)^{54}}{\left(x^{\frac{8}{5}}w^{16}\right)^{54}}$$

Expand \left(z^{\frac{2}{5}}y^{\frac{121}{15}}\right)^{54}.

$$\frac{\left(z^{\frac{2}{5}}\right)^{54}\left(y^{\frac{121}{15}}\right)^{54}}{\left(x^{\frac{8}{5}}w^{16}\right)^{54}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{2}{5} and 54 to get \frac{108}{5}.

$$\frac{z^{\frac{108}{5}}\left(y^{\frac{121}{15}}\right)^{54}}{\left(x^{\frac{8}{5}}w^{16}\right)^{54}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{121}{15} and 54 to get \frac{2178}{5}.

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{\left(x^{\frac{8}{5}}w^{16}\right)^{54}}$$

Expand \left(x^{\frac{8}{5}}w^{16}\right)^{54}.

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{\left(x^{\frac{8}{5}}\right)^{54}\left(w^{16}\right)^{54}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{8}{5} and 54 to get \frac{432}{5}.

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{x^{\frac{432}{5}}\left(w^{16}\right)^{54}}$$

To raise a power to another power, multiply the exponents. Multiply 16 and 54 to get 864.

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{x^{\frac{432}{5}}w^{864}}$$

Expand

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{x^{\frac{432}{5}}w^{864}}$$

Solution Steps

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

$$\left(\frac{\left(w^{-1}\right)^{\frac{3}{5}}\left(x^{3}\right)^{\frac{3}{5}}\left(z^{2}\right)^{\frac{3}{5}}\left(w^{3}x^{3}y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}\left(x^{3}\right)^{\frac{3}{5}}\left(z^{2}\right)^{\frac{3}{5}}\left(w^{3}x^{3}y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply 3 and \frac{3}{5} to get \frac{9}{5}.

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}\left(z^{2}\right)^{\frac{3}{5}}\left(w^{3}x^{3}y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}\left(w^{3}x^{3}y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}\left(w^{3}\right)^{\frac{1}{5}}\left(x^{3}\right)^{\frac{1}{5}}\left(y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}w^{\frac{3}{5}}\left(x^{3}\right)^{\frac{1}{5}}\left(y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}w^{\frac{3}{5}}x^{\frac{3}{5}}\left(y^{\frac{1}{3}}\right)^{\frac{1}{5}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and \frac{1}{5} to get \frac{1}{15}.

$$\left(\frac{w^{-\frac{3}{5}}x^{\frac{9}{5}}z^{\frac{6}{5}}w^{\frac{3}{5}}x^{\frac{3}{5}}y^{\frac{1}{15}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

Multiply w^{-\frac{3}{5}} and w^{\frac{3}{5}} to get 1.

$$\left(\frac{x^{\frac{9}{5}}z^{\frac{6}{5}}x^{\frac{3}{5}}y^{\frac{1}{15}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To multiply powers of the same base, add their exponents. Add \frac{9}{5} and \frac{3}{5} to get \frac{12}{5}.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{\left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

Expand \left(w^{-4}x^{-1}y^{2}z^{-\frac{1}{5}}\right)^{-4}.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{\left(w^{-4}\right)^{-4}\left(x^{-1}\right)^{-4}\left(y^{2}\right)^{-4}\left(z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply -4 and -4 to get 16.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{w^{16}\left(x^{-1}\right)^{-4}\left(y^{2}\right)^{-4}\left(z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -4 to get 4.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{w^{16}x^{4}\left(y^{2}\right)^{-4}\left(z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

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

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{w^{16}x^{4}y^{-8}\left(z^{-\frac{1}{5}}\right)^{-4}}\right)^{54}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{1}{5} and -4 to get \frac{4}{5}.

$$\left(\frac{x^{\frac{12}{5}}z^{\frac{6}{5}}y^{\frac{1}{15}}}{w^{16}x^{4}y^{-8}z^{\frac{4}{5}}}\right)^{54}$$

Cancel out z^{\frac{4}{5}}x^{\frac{12}{5}} in both numerator and denominator.

$$\left(\frac{\sqrt[15]{y}z^{\frac{2}{5}}}{y^{-8}x^{\frac{8}{5}}w^{16}}\right)^{54}$$

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

$$\left(\frac{z^{\frac{2}{5}}y^{\frac{121}{15}}}{x^{\frac{8}{5}}w^{16}}\right)^{54}$$

To raise \frac{z^{\frac{2}{5}}y^{\frac{121}{15}}}{x^{\frac{8}{5}}w^{16}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(z^{\frac{2}{5}}y^{\frac{121}{15}}\right)^{54}}{\left(x^{\frac{8}{5}}w^{16}\right)^{54}}$$

Expand \left(z^{\frac{2}{5}}y^{\frac{121}{15}}\right)^{54}.

$$\frac{\left(z^{\frac{2}{5}}\right)^{54}\left(y^{\frac{121}{15}}\right)^{54}}{\left(x^{\frac{8}{5}}w^{16}\right)^{54}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{2}{5} and 54 to get \frac{108}{5}.

$$\frac{z^{\frac{108}{5}}\left(y^{\frac{121}{15}}\right)^{54}}{\left(x^{\frac{8}{5}}w^{16}\right)^{54}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{121}{15} and 54 to get \frac{2178}{5}.

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{\left(x^{\frac{8}{5}}w^{16}\right)^{54}}$$

Expand \left(x^{\frac{8}{5}}w^{16}\right)^{54}.

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{\left(x^{\frac{8}{5}}\right)^{54}\left(w^{16}\right)^{54}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{8}{5} and 54 to get \frac{432}{5}.

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{x^{\frac{432}{5}}\left(w^{16}\right)^{54}}$$

To raise a power to another power, multiply the exponents. Multiply 16 and 54 to get 864.

$$\frac{z^{\frac{108}{5}}y^{\frac{2178}{5}}}{x^{\frac{432}{5}}w^{864}}$$