Question

Simplify 3 square root of 10 end root plus 7 square root of 15 end root minus 6 square root of 10 end root minus 4 square root of 15

Answer 1

Start with

$3\sqrt{10}+7\sqrt{15}-6\sqrt{10}-4\sqrt{15}$

We can factor the square root of 10 between the 1st and 3rd term, and the square root of 15 between the remaining two:

$\sqrt{10}(3-6)+\sqrt{15}(7-4) = -3\sqrt{10}+3\sqrt{15}$

Now we can factor 3 from both terms:

$3(\sqrt{15}-\sqrt{10})$

You can also use the rule

$\sqrt{ab}=\sqrt{a}\sqrt{b}$

To observe that

$3(\sqrt{15}-\sqrt{10})=3(\sqrt{5}\sqrt{3}-\sqrt{5}\sqrt{2})$

So we can factor the square root of 5 as well:

$3\sqrt{5}(\sqrt{3}-\sqrt{2})$

And the expression is fully simplified.