Step 1

The given equation is, \(\displaystyle{\left({x}^{{4}}{y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}{y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

Step 2

Assume that the abive equation is true for all positive real values of x and y.

Now obtain the values of constants j and k as shown below.

\(\displaystyle{\left({x}^{{4}}{y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}{y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

\(\displaystyle{\left({x}^{{4}}\right)}^{{\frac{{1}}{{4}}}}{\left({y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}\right)}^{{\frac{{1}}{{5}}}}{\left({y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

\(\displaystyle{\left({x}{y}^{{\frac{{5}}{{4}}}}\right)}{\left({x}^{{\frac{{8}}{{5}}}}\right)}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

\(\displaystyle{\left({x}^{{{1}+\frac{{8}}{{5}}}}\right)}{\left({y}^{{{1}+\frac{{5}}{{4}}}}\right)}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

\(\displaystyle{x}^{{\frac{{13}}{{5}}}}{y}^{{\frac{{9}}{{4}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

Step 3

Equate the powers and obtain the values of j and k as follows.

\(\displaystyle\frac{{j}}{{5}}=\frac{{13}}{{5}}\)

j=13

\(\displaystyle\frac{{k}}{{4}}=\frac{{9}}{{4}}\)

k=9

Now compute the difference j-k as shown below.

j-k=13-9

=4

Therefore, the correct option is B.

The given equation is, \(\displaystyle{\left({x}^{{4}}{y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}{y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

Step 2

Assume that the abive equation is true for all positive real values of x and y.

Now obtain the values of constants j and k as shown below.

\(\displaystyle{\left({x}^{{4}}{y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}{y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

\(\displaystyle{\left({x}^{{4}}\right)}^{{\frac{{1}}{{4}}}}{\left({y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}\right)}^{{\frac{{1}}{{5}}}}{\left({y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

\(\displaystyle{\left({x}{y}^{{\frac{{5}}{{4}}}}\right)}{\left({x}^{{\frac{{8}}{{5}}}}\right)}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

\(\displaystyle{\left({x}^{{{1}+\frac{{8}}{{5}}}}\right)}{\left({y}^{{{1}+\frac{{5}}{{4}}}}\right)}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

\(\displaystyle{x}^{{\frac{{13}}{{5}}}}{y}^{{\frac{{9}}{{4}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}\)

Step 3

Equate the powers and obtain the values of j and k as follows.

\(\displaystyle\frac{{j}}{{5}}=\frac{{13}}{{5}}\)

j=13

\(\displaystyle\frac{{k}}{{4}}=\frac{{9}}{{4}}\)

k=9

Now compute the difference j-k as shown below.

j-k=13-9

=4

Therefore, the correct option is B.