Question

How to compare the following numbers without using the calculator?
√5-√3 and √(8-2√15)

Answer 1

Answer on Question#37228

How to compare the following numbers without using the calculator?

$\sqrt{5} - \sqrt{3}$ and $\sqrt{8 - 2\sqrt{15}}$

Solution.

Consider the second number. Let's isolate a perfect square:

$$8 - 2 \sqrt {15} = 5 - 2 \sqrt {15} + 3 = \sqrt {5} ^ {2} - 2 \cdot \sqrt {5} \cdot \sqrt {3} + \sqrt {3} ^ {2} = (\sqrt {5} - \sqrt {3}) ^ {2}$$

So

$$\sqrt {8 - 2 \sqrt {15}} = \sqrt {\left(\sqrt {5} - \sqrt {3}\right) ^ {2}} = \left| \sqrt {5} - \sqrt {3} \right| = \sqrt {5} - \sqrt {3}$$

We can see that $\sqrt{5} - \sqrt{3} = \sqrt{8 - 2\sqrt{15}}$

Answer: $\sqrt{5} - \sqrt{3} = \sqrt{8 - 2\sqrt{15}}$