Question

6^{\frac{2}{x}}+4^{\frac{1}{x}}=\frac{10}{3}\cdot 12^{\frac{1}{x}}`

Answer 1

Answer:

x = 1

x = -1

Step-by-step explanation:

Solve the equation

$6^{\frac{2}{x}}+4^{\frac{1}{x}}=\dfrac{10}{3}\cdot 12^{\frac{1}{x}}$

First, note that

$4^{\frac{1}{x}}=2^{\frac{2}{x}}$

Now, divide the whole equation by $2^{\frac{2}{x}}$

$\dfrac{6^{\frac{2}{x}}}{2^{\frac{2}{x}}}+\dfrac{2^{\frac{2}{x}}}{2^{\frac{2}{x}}}=\dfrac{10}{3}\cdot \dfrac{12^{\frac{1}{x}}}{2^{\frac{2}{x}}}$

$3^{\frac{2}{x}}+1=\dfrac{10}{3}\cdot \dfrac{12^{\frac{1}{x}}}{4^{\frac{1}{x}}}\\ \\3^{\frac{2}{x}}+1=\dfrac{10}{3}\cdot 3^{\frac{1}{x}}$

Use substitution

$t=3^{\frac{1}{x}}$

Then

$t^2+1=\dfrac{10}{3}t$

Multiply by 3:

$3t^2+3=10t\\ \\3t^2-10t+3=0\\ \\D=(-10)^2-4\cdot 3\cdot 3=100-36=64\\ \\t_{1,2}=\dfrac{-(-10)\pm \sqrt{64}}{2\cdot 3}=\dfrac{10\pm 8}{6}=3,\dfrac{1}{3}$

Thus,

$3^{\frac{1}{x}}=3\Rightarrow \dfrac{1}{x}=1,\ x=1\\ \\3^{\frac{1}{x}}=\dfrac{1}{3}\Rightarrow \dfrac{1}{x}=-1,\ x=-1$

Answer 2

This is the right answer and how!

~!HOPE IT HELPS!~