Some questions about power as a whole.

Question

So, I live in a poor country, and therefore I got a really bad elementary and middle school. I am studying for a university and something that I really can't understand is the relation below.

$$\sqrt[3]{a^3 + b^3} = a + b$$

I know it is wrong, but I fail to understand why.

Thinking about it, my difficulty is probably related to the use of parentheses and it's meaning. For example, is $(a + b)^2$ different from $a^2 + b^2$? What happens if I apply a square root on them? I know that if $2x^2 = y^2$, I can apply the square root and get $2x = y$, but can I do the same to $2x^2 = y^2 + z^2$ and get $2x = y + z$? If not, why?

Sorry for such a trivial boring question, but I can't find answers to those questions anywhere else. I would really appreciate if you guys could answer them all.

Thanks!

Answer 1

Consider an example. Can we see that $\sqrt[3]{a^3+b^3}$ is not equal to $a+b$ in some simple case? How about the case $a=b=1$ ...

Let's show that $\sqrt[3]{1^3+1^3}$ is not equal to $1+1$.

First, compute $$ 1+1 = 2 $$

Next, note that $1^3 = 1 \times 1 \times 1 = 1$. Then $$ \sqrt[3]{1^3+1^3} = \sqrt[3]{1+1}=\sqrt[3]{2} $$ Some number whose cube is $2$. But this is not $2$, since $2$ does not have cube equal to $2$. Actually, $2$ has cube equal to $8$.

The cube of $\sqrt[3]{1^3+1^3}$ is $2$.
The cube of $1+1$ is $8$.
So the two numbers $\sqrt[3]{1^3+1^3}$ and $1+1$ are different.

Answer 2

I'd just like to add one more thing to GEdgar's answer.

Another way to see they are not equal is to work backwards. Start out with $a+b$. We will raise this to the third power.

$$(a+b)^3 = (a+b)^2(a+b) = (a^2 + 2ab +b^2) (a+b ) = a^3 + 3a^2b + 3ab^2 + b^3$$

Now on the other hand, your original equation claims that $\sqrt[3]{a^3+b^3} = a+b$, which means that cubing both sides of this equation, we would have $a^3 + b^3 = a^3 + 3a^2b + 3ab^2 + 3b^3$. But this is clearly not true. For example, if $a$ and $b$ are positive, the right side is greater than the left side.