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Does $x \times \sqrt x = x$?

I thought it was correct because sqrt is the opposite of multiplying by a number, so I figured by multiplying by a number it would balance out and be that number normally, but when I tried it with my Python calculator using 3 I got:

math.sqrt(3) * 3
5.196152422706632

qwr
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taz henery duck0
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    You seem very confused... If it were true that $x\sqrt{x} = x$ for all $x$, that would imply $\sqrt{x} = 1$ for all $x$. The "opposite of multplying", if anything, is dividing, not taking square roots. – fkraiem Feb 02 '14 at 01:41
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    $\sqrt x$ is the number satisfying $\sqrt x \sqrt x = x$; it's not the opposite of multiplying by $x$. –  Feb 02 '14 at 01:42
  • Squareroot is the opposite of squaring. One can say for $x>0$ that $\sqrt{x}^2=x$ and that $\sqrt{x^2}=x.$ – coffeemath Feb 02 '14 at 01:43
  • The inverse function of multiplication is division, while the inverse function of squaring a number is taking the square root of the result. – Yiyuan Lee Feb 02 '14 at 01:47
  • Also, why bother with Python? $\sqrt 4 = 2$, but $4 \cdot 2 \ne 4$. –  Feb 02 '14 at 01:47
  • @fkraiem Certainly, it would imply $\sqrt{x}=1$ or $x=0$. – mathematics2x2life Feb 02 '14 at 01:56
  • @T.Bongers Python is fun? I dunno – qwr Feb 02 '14 at 01:57

4 Answers4

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In general, no. Since $\sqrt x$ is equal to $x^{1/2}$, your equation is the same as $x^{3/2} = x$, only $x = 0, 1$ work as solutions

qwr
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If you want to solve for the equation $$ x\sqrt{x} = x $$ then you have $$ x (\sqrt{x}-1) =0 \Rightarrow x = 0, \text{ or } \sqrt{x}=1, \text{ i.e. } x = 1 $$

user44197
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The multiplicative inverse (the "opposite" in your question) of a non-zero number $x$ is its reciprocal, $\frac{1}{x}$. Multiplying these two together gives instead $x \times \frac{1}{x} = 1$.

Yiyuan Lee
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so basically $\sqrt x$ is the same as $x^{1/2}$ and x can also be taken as $x^1$ that means when you multiply both of them you'll get $x^{1+1/2}$ which would be $x^{3/2}$

Christian Barajas
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