Why can't we remove the radicals from $ \sqrt a + \sqrt b = 8 $ by writing $(\sqrt a)^2 + (\sqrt b)^2 = 8^2$?

Question

Given the equation:

$$ \sqrt a + \sqrt b = 8 $$

Why is it wrong to remove the radicals like this?

$$\sqrt a + \sqrt b = 8 $$

$$(\sqrt a)^2 + (\sqrt b)^2 = 8^2$$

$$a + b = 64$$

Instead you have to do this way as my answer key says:

$$(\sqrt a + \sqrt b)^2 = 8^2 $$

$$a + 2\sqrt ab + b = 64$$

Doesn't the first method suffice in terms of doubling both sides?

Answer 1

You can't remove radicals that way because it results in false equations.

Let's try it with an explicit example: $$\sqrt{4} + \sqrt{9} = 5 $$ Fine so far. Now let's try what you suggest: $$(\sqrt{4})^2 + (\sqrt{9})^2 = 4 + 9 = 13 \ne 25 = 5^2 $$

In general, you can always test a proposed "law of algebra" by plugging in numbers. All you have to do is to find one counterexample which makes the law false, and the proposed "law" falls apart.

Answer 2

The first method is wrong because: if $a+b=c$, it does not follow that $a^2+b^2=c^2$. For example, $1+1=2$, but $1^2+1^2 \neq 2^2$.

Answer 3

As a general rule, if you have an equality and you wish to algebraically manipulate it, you may apply a function to both sides. Such an application must be done to each side as a whole rather than to each individual part.

For example (using a messy left-hand side for illustrative purposes)

Suppose that we are told that $a+a\cdot b + b^2 = c$

In that case, we can conclude that $(a+a\cdot b + b^2) + 2 = (c) + 2$ by applying the function of "adding two."

Similarly, we could conclude that $(a+a\cdot b + b^2)\cdot 2 = (c)\cdot 2$ by applying the function of "multiplying by two."

We could conclude that $\sin(e^{(a+a\cdot b + b^2)})=\sin(e^{(c)})$ by first taking $e$ to the power of each side and then taking the sine of each side, etc...

Note that in all of the above, even if we originally wrote the sides of the original equality without parentheses, it helps to visualize exactly what is going on by putting them in parentheses first before making the change.

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For your specific example, you began with $\sqrt{a}+\sqrt{b}=8$. We went to algebraically manipulate each side by squaring each.

This leads us to $(\sqrt{a}+\sqrt{b})^2=(8)^2$.

We then remember how squaring of a number works: we multiply it by itself. This leads us to $(\sqrt{a}+\sqrt{b})\times (\sqrt{a}+\sqrt{b})=8\times 8$

We can then apply distributivity laws on one set of parentheses at a time, or "FOIL" the expression, eventually leading us to $a+2\sqrt{ab}+b=64$.