Why isn't $(2x+x^2)^{1/2}$ the same as $(2x)^{1/2}+x$?

Question

I just don't get why this isn't true.

Answer 1

Taking a power of something isn't a linear operation

Answer 2

You have $$ \sqrt{2x + x^2} = \sqrt{x(2+x)} = \sqrt{x} \sqrt{2+x} $$ but in general, $$ \sqrt{a+b} \ne \sqrt{a} + \sqrt{b}. $$ Suppose this "equality" indeed held, then squaring both sides yields $$ a+b = a + b + 2\sqrt{ab} \iff \sqrt{ab} = 0 $$ So you can see either $a=0$ or $b=0$ are needed for your "equality" to hold...

Answer 3

Maybe a picture convinces you.

Answer 4

let $y=(2x+x^2)^{0.5}$. And $z=(2x)^{0.5} +x$. Then $y^2=2x+x^2$ and $z^2 =((2x)^{0.5}+x)^{2}$. Then it should be clear that $z$ can not be equal to $y$ since the well defined operations do not match. Therefore, you can not define it that way.

Answer 5

Why should it be true. other than wishful thinking? When you first learn algebra you remember the distributive law for "multiplication by a fixed constant $a$", namely $$ a(x + y) = ax +ay . $$ That is true because you can prove it, first for integers and then for any real numbers.

But, sadly, $$ (x + y)^2 \ne x^2 + y ^2 \\ $$ $$ \sqrt{x + y} \ne \sqrt{x} + \sqrt{y} $$ $$ a^{x + y} \ne a^{x} + a^{y} $$ $$ \log(x + y) \ne \log{x} + \log{y} $$ $$ \sin(x + y) \ne \sin{x} + \sin{y} $$

See

• Pedagogy: How to cure students of the "law of universal linearity"?
• Why is the square root of a sum not equal to the square root of each its addends?

Answer 6

Square both, you see:

$$(\sqrt{2x+x^2})^2\equiv(\sqrt{2x}+x)^2$$

$$2x+x^2\equiv(\sqrt{2x}+x)(\sqrt{2x}+x)\equiv2x+x^2+2x\sqrt{2x}$$ $$\implies 0\equiv2x\sqrt 2x$$ which is clearly false.

Answer 7

This is (a version of) the "freshman's binomial". While it is true in characteristic $p$ that $(x+y)^p=x^p+y^p$, this is false in general.

Answer 8

Hint: Test the equation with $x=1$. You will see that your equations give different results. The square root is the inverse of the square operator. It is clear that $(a+b)^2\neq a^2 + b^2$ hence the inversion $\sqrt{a^2+b^2}$ is in general not equal to $|a+b|$.