kuCAu

2021-03-08

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

pattererX

Step 1
The given equation is, ${\left({x}^{4}{y}^{5}\right)}^{\frac{1}{4}}{\left({x}^{8}{y}^{5}\right)}^{\frac{1}{5}}={x}^{\frac{j}{5}}{y}^{\frac{k}{4}}$
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.
${\left({x}^{4}{y}^{5}\right)}^{\frac{1}{4}}{\left({x}^{8}{y}^{5}\right)}^{\frac{1}{5}}={x}^{\frac{j}{5}}{y}^{\frac{k}{4}}$
${\left({x}^{4}\right)}^{\frac{1}{4}}{\left({y}^{5}\right)}^{\frac{1}{4}}{\left({x}^{8}\right)}^{\frac{1}{5}}{\left({y}^{5}\right)}^{\frac{1}{5}}={x}^{\frac{j}{5}}{y}^{\frac{k}{4}}$
$\left(x{y}^{\frac{5}{4}}\right)\left({x}^{\frac{8}{5}}\right)={x}^{\frac{j}{5}}{y}^{\frac{k}{4}}$
$\left({x}^{1+\frac{8}{5}}\right)\left({y}^{1+\frac{5}{4}}\right)={x}^{\frac{j}{5}}{y}^{\frac{k}{4}}$
${x}^{\frac{13}{5}}{y}^{\frac{9}{4}}={x}^{\frac{j}{5}}{y}^{\frac{k}{4}}$
Step 3
Equate the powers and obtain the values of j and k as follows.
$\frac{j}{5}=\frac{13}{5}$
$j=13$
$\frac{k}{4}=\frac{9}{4}$
$k=9$
Now compute the difference $j-k$ as shown below.
$j-k=13-9$
$=4$

Therefore, the correct option is B.

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