Question

2√6-√20 / 2√5+√24 * (11+√30)

Answer 1

$\frac{(2 \sqrt{6} - 2 \sqrt{5} )(2 \sqrt{5 } - 2 \sqrt{6)} }{(2 \sqrt{5} + 2 \sqrt{6})(2 \sqrt{5} - 2 \sqrt{6} ) } \times$
$\frac{ - (20 - 8 \sqrt{30} + 24)}{24 - 20} =$
$\frac{ - 4(11 - \sqrt{30}) }{4} =$
$( \sqrt{30} - 11)(11 + \sqrt{30} ) = 121 - 30 = 91$

Answer 2

Let's simplify the expression step by step:

Step 1: Simplify the individual square roots inside the expression.

√6 = √(2 * 3) = √2 * √3 √20 = √(2 * 2 * 5) = √4 * √5 = 2√5 √5 = √5 √24 = √(2 * 2 * 2 * 3) = √(8 * 3) = √8 * √3 = 2√2 * √3 √30 = √(2 * 3 * 5) = √6 * √5

Step 2: Replace the square roots with their simplified forms in the expression.

(2√6 - √20) / (2√5 + √24) * (11 + √30) = (2(√2 * √3) - 2√5) / (2√5 + 2√2 * √3) * (11 + √6 * √5) = (2√2 * √3 - 2√5) / (2√5 + 2√2 * √3) * (11 + √6 * √5)

Step 3: Combine like terms in the numerator and denominator.

Common factor in the numerator: 2√2 * √3 - 2√5 = 2(√2 * √3 - √5) Common factor in the denominator: 2√2 * √3 + 2√5 = 2(√2 * √3 + √5)

(2(√2 * √3 - √5)) / (2(√2 * √3 + √5)) * (11 + √6 * √5)

Step 4: Cancel out the common factors in the numerator and denominator.

The 2's and (√2 * √3) will cancel out:

(√2 * √3 - √5) / (√2 * √3 + √5) * (11 + √6 * √5)

Step 5: Apply the difference of squares formula to simplify further.

The difference of squares formula states that (a^2 - b^2) = (a - b)(a + b).

In our case, a = √2 * √3 and b = √5:

(√2 * √3 - √5) = (√2 * √3 - √5) * (1) = (√2 * √3 - √5) * (√2 * √3 + √5) / (√2 * √3 + √5)

Step 6: Cancel out the common factors in the numerator and denominator again.

(√2 * √3 - √5) / (√2 * √3 + √5) * (√2 * √3 + √5) / (√2 * √3 + √5)

Step 7: Simplify further.

The (√2 * √3 + √5) in the numerator cancels with the (√2 * √3 + √5) in the denominator:

Result = √2 * √3 - √5

Finally, we cannot simplify this expression any further. Thus, the simplified form of the given expression is √2 * √3 - √5.