solve for x, giving answer to 3s.f?

Question

I need help solving the question below:

$$ 2x^ \frac{1}{4} = \frac {64} {x} $$

I know the answer is 16 but I'm not sure how to get to it. Can you explain how to get the answer so I can solve similar questions

Answer 1

$$2x^{\dfrac14}\cdot x=32\implies x^{\left(\dfrac14+1\right)}=32=2^5\implies x^{\dfrac54}=2^5$$

$$\implies x^{\dfrac14}=(2^5)^{\dfrac15}$$

Now one of the five values of $\displaystyle(2^5)^{\dfrac15}$ is $2$

In that case $\displaystyle x^{\dfrac14}=2\implies (x^{\dfrac14})^4=2^4\implies x=2^4$

Answer 2

$2x^{\frac{1}{4}}(x) = 64 \Rightarrow 2x^\frac{1}{4}x^\frac{4}{4} = 64 \Rightarrow 2x^\frac{5}{4} = 64 \Rightarrow x^{\frac{5}{4}}=32 \Rightarrow x =(32)^{\frac{4}{5}}$

As, $2^5=32 \Rightarrow x = (2^5)^{\frac{4}{5}} = 2^4 = 16$

Answer 3

Multiply both sides by $x$, and you get $$2x^{\frac{5}{4}}=64$$ and then $$x^{\frac{5}{4}}=32$$ $$x^{\frac{5}{4}}=2^5$$ $$x^{\frac{1}{4}}=2$$ $$x=2^4=16$$

Answer 4

$x$ is positive, so you can take the logarithm and get $\ln2+\frac14\ln x=\ln(64)-\ln x$ and solve for $\ln x$. Since $\ln(64)=6\ln 2$, you get easily to $\ln x=4\ln 2$. $x=16$ follows after taking the exponential.