8
$\begingroup$

The following question is a Math Olympiad Problem:

Find $x>0$ that solves $x\sqrt{x\sqrt x}=2$

The answer is $\sqrt[7]{16}$ but I got $2\sqrt[3]{2}$ by: $$\begin{align}x\sqrt{x\sqrt x}=2&\to \sqrt{x\sqrt{x}}=\frac2x\\&\to x\sqrt{x}=\frac{4}{x^{2}}\\&\to \sqrt{x}=\frac{4}{x}\to x=\frac{16}{x^{2}}\\&\to x^{3}=16\\&\to x=\sqrt[3]{16}\\&\to x=2\sqrt[3]2\end{align}$$

What have I done wrong?

$\endgroup$
5
  • 6
    $\begingroup$ Welcome to Math SE. One issue is going from $x\sqrt{x}=\frac{4}{x^2}$ to your $\sqrt{x}=\frac{4}{x}$. You apparently divided both sides by $x$ to get rid of the $x$ on the LHS, so the result should then be $\sqrt{x}=\frac{4}{x^{\color{red}{3}}}$ instead. $\endgroup$
    – John Omielan
    Commented Apr 23 at 18:04
  • 3
    $\begingroup$ Do you mean $\sqrt[7]{16}$? $\endgroup$
    – Sai Mehta
    Commented Apr 23 at 18:05
  • 1
    $\begingroup$ You were fine up until $$x\sqrt{x}=\frac{4}{x^2},$$ then you made a mistake. $\endgroup$
    – Cameron Buie
    Commented Apr 23 at 18:05
  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Apr 23 at 18:09
  • $\begingroup$ Better solve reworking the exponents: $x^{1+(1+1/2)/2}=x^{7/4}=2$. $\endgroup$
    – Yves Daoust
    Commented Aug 12 at 6:47

2 Answers 2

Reset to default
14
$\begingroup$

$$\color{red}{x\sqrt{x}=\frac{4}{x^{2}}}\to \color{purple}{\sqrt{x}=\frac{4}{x}}$$

I think you can understand your mistake.

$\endgroup$
1
  • $\begingroup$ Thanks, found my error, i guess my mind isnt working today. $\endgroup$
    – Kcharliee
    Commented Apr 23 at 18:15
14
$\begingroup$

The answer by MathStackexchangeIsMarvellous identifies the error in the method that you have used.

Here is a simpler alternative approach to the one you have taken which uses the rules of exponentiation to simplify the problem:

$$ x \sqrt{x \sqrt{x}} = x \sqrt{x \cdot x^{1/2}} = x \sqrt{x^{3/2}} = x(x^{3/2})^{1/2}$$ $$= x \cdot x^{3/4} = x^{7/4}$$

So we substitute the above into your problem as follows:

$$ x \sqrt{x \sqrt{x}} = 2 \iff x^{7/4} = 2 \iff x^7 = 16$$

Therefore, we find that the solution is $x= \sqrt[7]{16}$, as required.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .