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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!

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    Actually $2x^2=y^2$ then you have $\sqrt{2}x=y$ assuming $x,y>0$.Also try multiplying $(a+b)\cdot (a+b)$ is the result $a^2+b^2$?Try multiplying $(a+b)\cdot(a+b)\cdot(a+b)$ is that $a^3+b^3$? – 2017-02-13

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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.

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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.

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    So $a + b$ is the same of $(a + b)$? – 2017-02-13
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    @ArthurCuesta Yes. The only reason I need parentheses is to point out that I want to raise the whole expression to a power. – 2017-02-13
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    You are getting the right idea here. Take note that parentheses are used to clarify the order of operations. For example, $2 + 3^2$ we have a basic rule that exponents are done ahead of addition -- people memorize an order of operation rule by BEDMAS (Brackets, Of, Division and Multiplication, Addition and Subtraction) or other mnemonics.So $2 + 3^2 = 2 + 9 = 11$ But $(2+3)^2$ means add *first* (Brackets or Parentheses) so $(2+3)^2 = 5^2 = 25.$ Thus the brackets clarify the meaning. Note also $3^2 + 4^2 = 9 + 16 = 25$ so $\sqrt(3^2 + 4^2) = \sqrt(25) = 5$ But 3 + 4 = 7, not 5. – 2017-02-14
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    One good way to improve your math is to buy used textbooks with answers at the back and work through them. I recommend Schaum's Guides if you can get your hands on some. They are inexpensive new and very affordable used. You would find the Algebra guide useful. Other schoolbooks can also help. Age is no problem for basic math, which does not change. By university level newer books help. – 2017-02-14
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    This is actually really interesting. For example: $\sqrt[3]{a^3 + b^3} = a + b$. Putting all to the third power: $a^3 + b^3 = a^3 + 3a^2b + 3ab^2 + b^3$. Why doesnt the left side of the equation also becomes $a^3 + 3a^2b + 3ab^2 + b^3$? Math is really weird. – 2017-02-14
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    Anyway, thanks for the tips. – 2017-02-14