# I don't understand how to factor the expressions $\sqrt{57+40\sqrt2} - \sqrt{57-40\sqrt2}$ [closed]

$$\sqrt{57+40\sqrt2} - \sqrt{57-40\sqrt2}$$

I'm completely clueless about how to use the formula $$a^2 + 2ab + b^2 = (a+b)^2$$ to factor the expressions.

With the help of commenters, I successfully factored the expressions.

$$\sqrt{57+40\sqrt{2}} = 5+4\sqrt{2}$$ and $$\sqrt{57-40\sqrt{2}} = 5-4\sqrt{2}$$. $$5+4\sqrt{2}-4\sqrt{2}+5=10$$.

• $$(a+b\sqrt2)^2 = a^2+2ab\sqrt2+2b^2 = (a^2+2b^2) +2ab\sqrt2.$$ Can you find two numbers $a$ and $b$ such that $a^2+2b^2=57$ and $2ab=40$? Commented Sep 11, 2022 at 8:40
• If you are trying to simplify $x=\sqrt{57+40\sqrt2} - \sqrt{57-40\sqrt2}$, you can try to factor $57+40\sqrt{2}$ as @Rócherz explained. Alternatively, you can compute $x^2$ and use identities. Commented Sep 11, 2022 at 9:32
• Welcome to MathSE. The reason your question has received down votes is that you have not included your attempt. This tutorial explains how to typeset mathematics on this site. Commented Sep 11, 2022 at 9:46
• For future reference, you might want to read our guidelines for how to avoid "I'm completely clueless" questions. Commented Sep 11, 2022 at 13:14

$$\sqrt{57+40\sqrt{2}}=\sqrt{25+40\sqrt{2}+32}=\sqrt{5^2+2\cdot 5\cdot 4\sqrt{2}+(4\sqrt{2})^2}=\sqrt{(5+4\sqrt{2})^2}=5+4\sqrt{2}$$
In general, when asked to simplify an expression of the form $$z:=\sqrt{x+a\surd y}\pm\sqrt{x-a\surd y},$$ the trick is to square it, and see what happens. Thus $$z^2=\left(\sqrt{x+a\surd y}\pm\sqrt{x-a\surd y}\right)^2$$, and so $$z^2=2x\pm2\sqrt{x^2-a^2y}.$$ As usual in this type of problem, $$x^2-a^2y$$ is a perfect square: in this case, $$57^2-40^2\cdot2=49=7^2$$. After taking the square root, we end up with $$z=\sqrt{2\cdot57-2\cdot7}=10.$$
$$\sqrt{57+40\sqrt2} - \sqrt{57-40\sqrt2}$$ $$\Rightarrow \sqrt{25+32+2.(5).(4\sqrt2}) - \sqrt{25+32-2.(5).(4\sqrt2)}$$ $$\Rightarrow 5+4\sqrt2 - (4\sqrt2-5)$$ $$\Rightarrow 10$$
• Your answer is incorrect. $5 - 4\sqrt{2} = \sqrt{25} - \sqrt{32} < 0$. Therefore, it cannot be the square root of $\sqrt{57 - 40\sqrt{2}}$. Commented Sep 11, 2022 at 12:53