Question

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

Answer 1

This is the right answer and how!

~!HOPE IT HELPS!~

Answer 2

x = 1

x = -1

Explanation:

Solve the equation

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

First, note that

`4^{(1)/(x)}=2^{(2)/(x)}`

Now, divide the whole equation by
`2^{(2)/(x)}`

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

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

Use substitution

`t=3^{(1)/(x)}`

Then

`t^2+1=(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)=(-(-10)\pm √(64))/(2\cdot 3)=(10\pm 8)/(6)=3,(1)/(3)`

Thus,

`3^{(1)/(x)}=3\Rightarrow (1)/(x)=1,\ x=1\\ \\3^{(1)/(x)}=(1)/(3)\Rightarrow (1)/(x)=-1,\ x=-1`