1. ## Surds ii

I got (i) simplified to 12 + sq.rt35 + sq.rt35

(ii) How would you know if it is irrational without a calculator?

2. $(\sqrt{7}+\sqrt{5})^2=7+5+2\sqrt{7}\cdot\sqrt{5}=1 2+2\sqrt{35}$.
Suppose that $12+2\sqrt{35}\in\mathbf{Q}$.
Let $q\in\mathbf{Q}$ such as $12+2\sqrt{35}=q\Rightarrow \sqrt{35}=\frac{q-12}{2}\in\mathbf{Q}\Rightarrow \sqrt{35}\in\mathbf{Q}$, false.
So $12+2\sqrt{35}\not\in\mathbf{Q}$.

3. Originally Posted by red_dog
$(\sqrt{7}+\sqrt{5})^2=7+5+2\sqrt{7}\cdot\sqrt{5}=1 2+2\sqrt{35}$.
Suppose that $12+2\sqrt{35}\in\mathbf{Q}$.
Let $q\in\mathbf{Q}$ such as $12+2\sqrt{35}=q\Rightarrow \sqrt{35}=\frac{q-12}{2}\in\mathbf{Q}\Rightarrow \sqrt{35}\in\mathbf{Q}$, false.
So $12+2\sqrt{35}\not\in\mathbf{Q}$.

I don't understand this. What do the signs like the Q mean? And that fancy E means same as right?

I don't understand this. What do the signs like the Q mean? And that fancy E means same as right?
The "Q" means "rational" as in a fraction.

And $\in$ means "in", it means the number is "in" the rationals, i.e. the number is a fraction.

5. I'm sorry, I still don't understand how to work out if it is irrational or not without a calculator.

If the number under the square root sign is a perfect square, then it is rational. If it is not a perfect square then it is irrational. So anything with a $\sqrt{35}$ in it is going to be irrational.