Question

(sqrt(sqrt(10)-2)*sqrt(sqrt(10)+2))/sqrt(24)

Answer 1

$\frac{(\sqrt{\sqrt{10}-2}*\sqrt{\sqrt{10}+2})}{\sqrt{24}} = \frac{\sqrt6}{\sqrt{24}} = \sqrt\frac{1}{4}$

Answer 2

We can simplify the given expression by rationalizing the denominator as follows:

First, let's simplify the terms inside the square roots:

sqrt(sqrt(10)-2) can be written as (10^(1/4)-2)^(1/2)

sqrt(sqrt(10)+2) can be written as (10^(1/4)+2)^(1/2)

Now, let's multiply the numerator and denominator by (10^(1/4)+2)^(1/2), which is the conjugate of the numerator:

[(10^(1/4)-2)^(1/2) * (10^(1/4)+2)^(1/2)] / [(24)^(1/2) * (10^(1/4)+2)^(1/2)]

Simplifying the numerator using the difference of squares formula, we get:

[(10^(1/2) - 4)] / [(24)^(1/2) * (10^(1/4)+2)^(1/2)]

Now, we can simplify the denominator by factoring out 4 from the radicand:

[(10^(1/2) - 4)] / [(4*6)^(1/2) * (10^(1/4)+2)^(1/2)]

Simplifying further, we get:

[(10^(1/2) - 4)] / [(2*3)^(1/2) * (10^(1/4)+2)^(1/2)]

Now, we can simplify the numerator by multiplying both numerator and denominator by the conjugate of the numerator:

[(10^(1/2) - 4) * (10^(1/2) + 4)] / [(2*3)^(1/2) * (10^(1/4)+2)^(1/2) * (10^(1/2) + 4)]

Simplifying further, we get:

[(10 - 16)] / [(2*3)^(1/2) * (10^(1/4)+2)^(1/2) * (10^(1/2) + 4)]

Simplifying the numerator, we get:

[-6] / [(2*3)^(1/2) * (10^(1/4)+2)^(1/2) * (10^(1/2) + 4)]

Simplifying the denominator, we get:

[-6] / [(23)^(1/2) * (10^(1/4)+2)^(1/2) * (2(5)^(1/2) + 4)]

Finally, simplifying the denominator, we get:

[-6] / [2*(3*5)^(1/2) * (10^(1/4)+2)^(1/2)]

Simplifying the expression further, we get:

[-3*(5)^(1/2)] / [(352)^(1/2) * (10^(1/4)+2)^(1/2)]

Therefore, the simplified expression is:

[-3*(5)^(1/2)] / [(6*5)^(1/2) * (10^(1/4)+2)^(1/2)]