Fully simplify sqrt(49 - 20*sqrt(6)).

Answer 1

sqrt(49 - 20*sqrt(6))

First, let's simplify the expression using the given formula:

sqrt(a ± sqrt(b)) = √[(a ± √b)/2] ± √[(a ± √b)/2]

In this case, a = 49 and b = 6.

√(49 - 20*sqrt(6)) = √[(49 - √6)/2] ± √[(49 - √6)/2]

= √[(49 - √6)/2] - √[(49 - √6)/2]

Since we are taking the square root of another square root, we need to simplify further. Let's rewrite the expression a bit:

√[(49 - √6)/2] - √[(49 - √6)/2]
= √[(49 - 2√6 + 6)/2] - √[(49 - 2√6 + 6)/2]
= √[(55 - 2√6)/2] - √[(55 - 2√6)/2]
= (55 - 2√6)/2 - (55 - 2√6)/2
= 55/2 - 2√6/2 - 55/2 + 2√6/2
= 0

Therefore, the fully simplified form of sqrt(49 - 20*sqrt(6)) is 0.

Answer 2

clearly, 0 is wrong, since 20√6 is not 49
√(49-20√6) = √(25 - 20√6 + 24) = √(25 - 20√6 + (2√6)^2) = 5 - 2√6

Answer 3

You are correct. I apologize for the mistake in the previous response.

Let's simplify it correctly this time:

Given √(49 - 20√6)

We can rewrite 49 as 25 + 24:

√(25 - 20√6 + 24)

Next, we notice that -20√6 can be represented as -2√6 * 2√6:

√(25 - 2√6 * 2√6 + 24)

This can be represented as a perfect square of (5 - 2√6)^2:

√((5 - 2√6)^2)

Taking the square root of a perfect square cancels out the square:

5 - 2√6

Therefore, the fully simplified form of √(49 - 20√6) is 5 - 2√6. Thank you for pointing out the mistake.