Square Root of 5 plus 2 Square root of 2 over 4 minus the square root of 5 simplify

[ 5^.5 + 2 * 2^.5 ]/[4 - 5^.5]

multiply top and bottom by [4 + 5^.5]

[ 4*5^.5 + 5 +8*2^.5 + 2*10^.5]/11

To simplify the expression:

Square Root of 5 + 2 Square Root of 2
-----------------------------------
4 - Square Root of 5

We can start by simplifying the numerator and denominator separately.

Numerator: Square Root of 5 + 2 Square Root of 2

There are no like terms to combine, so we leave it as is.

Denominator: 4 - Square Root of 5

To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is 4 + Square Root of 5.

(4 + Square Root of 5)(4 + Square Root of 5) = 16 + 8 Square Root of 5 + 8 Square Root of 5 + 5 = 21 + 16 Square Root of 5

So, the rationalized denominator becomes 21 + 16 Square Root of 5.

Now, let's rewrite the expression with the simplified numerator and rationalized denominator:

(Square Root of 5 + 2 Square Root of 2) / (4 - Square Root of 5)

Becomes:

(Square Root of 5 + 2 Square Root of 2) / (21 + 16 Square Root of 5)

And that is the simplified form of the expression.

To simplify the expression (sqrt(5) + 2sqrt(2)) / (4 - sqrt(5)), we will use a technique called rationalizing the denominator.

Step 1: Rationalize the denominator
To remove the square root from the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (4 + sqrt(5)). Multiplying the numerator and denominator by this conjugate will help eliminate the square root from the denominator.

(sqrt(5) + 2sqrt(2)) / (4 - sqrt(5)) * (4 + sqrt(5)) / (4 + sqrt(5))

Step 2: Simplify the expression
Using the distributive property, multiply the numerator and denominator by the conjugate:

[(sqrt(5) + 2sqrt(2))(4 + sqrt(5))]/[(4 - sqrt(5))(4 + sqrt(5))]

Now, we can simplify the expression by applying the distributive property and combining like terms in the numerator:

[4sqrt(5) + sqrt(5)sqrt(5) + 8sqrt(2) + 2sqrt(2)sqrt(5)] / [(4^2 - (sqrt(5))^2)]

Simplifying further, we get:

(4sqrt(5) + 5 + 8sqrt(2) + 2sqrt(10))/(16 - 5)

(4sqrt(5) + 5 + 8sqrt(2) + 2sqrt(10))/11

So, the simplified expression is (4sqrt(5) + 5 + 8sqrt(2) + 2sqrt(10))/11