Question

# (x^4y^5)^(1/4)(x^8y^5)^(1/5)=x^(j/5)y^(k/4)In the equation above, j and k are constants. If the equation is true for all positive real values of x and y, what is the value of j - k?A)3B)4C)5D)6

Upper Level Math

$$\displaystyle{\left({x}^{{4}}{y}^{{5}}\right)}^{{\frac{{1}}{{4}}}}{\left({x}^{{8}}{y}^{{5}}\right)}^{{\frac{{1}}{{5}}}}={x}^{{\frac{{j}}{{5}}}}{y}^{{\frac{{k}}{{4}}}}$$
In the equation above, j and k are constants. If the equation is true for all positive real values of x and y, what is the value of $$j - k$$?
A)3
B)4
C)5
D)6

2021-03-09

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.