Match each expression with an equivalent one. In this question $x,y>1.$

$-vvy=3^x$

$-vv{e}^y=\frac{e^x}{e^3}{x}^y=3$

$y=e^{x}y=x{e}^{x-y}=3$

$y=x-3$

$y=\frac{ln(3)}{ln(x)}$

$y=e^{xln(3)}$

$x=ln(y)$

$y=ln(e^x)$

$y=x-ln(3)$

Let $x>0,$ we want to eliminate the square roots in the denominator of the expression

$\frac{\sqrt[\phantom{2}]{6}+3\sqrt[\phantom{2}]{x}}{\sqrt[\phantom{2}]{11}-\sqrt[\phantom{2}]{x}}$

a$)$ To reach our goal, we need to multiply the numerator and denominator of this fraction by what factor?

Answer:

$$

FORMATTING: In Mobius, we write $\sqrt[\phantom{2}]{x}$ as $\sqrt[\phantom{2}]{x}.$

b$)$ After multiplying and dividing by your answer in $($a$),$ and then simplifying, you will obtain a new expression of the form

$\frac{\sqrt[\phantom{2}]{6}+3\sqrt[\phantom{2}]{x}}{\sqrt[\phantom{2}]{11}-\sqrt[\phantom{2}]{x}}=\frac{Ax+B\sqrt[\phantom{2}]{x}+C}{D-x}$

where $A,B,C$ and $D$ are real numbers.

What are the exact values of $A,B,C$ and $D?$

$A=$

$B=(2)$

$C=$

$D=$

FORMATTING: In Mobius, we write $\sqrt[\phantom{2}]{x}$ as $\sqrt[\phantom{2}]{x}.$