Rationalize the denominator. The twos are outside of the square root signs.
5/(sqrt14)-2 x (sqrt14)+2/(sqrt14)+2
The answer is sqrt14+2/2 I am having trouible understanding to get rid of the 5
I figured it out after staring at what steps I already had written down.
To rationalize the denominator in the expression, we need to simplify the expression so that there are no square roots in the denominator. Let's break down the steps:
First, let's simplify the expression by combining like terms:
(5/(sqrt14))-2 x (sqrt14)+2/(sqrt14)+2
Next, we can simplify the expression by combining the terms involving the square root of 14:
(5/(sqrt14)) - 2(sqrt14) + 2/(sqrt14) + 2
Now, we need to focus on rationalizing the denominator of the fraction 5/(sqrt14). To do this, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (sqrt14). Multiplying the numerator by (sqrt14) gives us:
[5/(sqrt14)] x (sqrt14) = 5(sqrt14)/(sqrt14) = 5(sqrt14*sqrt14)/sqrt14 = 5(sqrt196)/sqrt14 = 5(14)/sqrt14 = 70/sqrt14
So, the expression becomes:
70/sqrt14 - 2(sqrt14) + 2/(sqrt14) + 2
Now, let's simplify further by combining the square root terms:
70/sqrt14 - 2(sqrt14) + 2/(sqrt14) + 2 = 70/sqrt14 - 2(sqrt14) + 2(sqrt14)/sqrt14 + 2
Since we have the same denominator (sqrt14) for the first and second terms, we can combine them:
70/sqrt14 - 2(sqrt14) = (70 - 2(sqrt14))/sqrt14
We can simplify the second and third terms, as follows:
2(sqrt14)/sqrt14 = 2
2/(sqrt14) = 2/(sqrt14) x (sqrt14)/(sqrt14) = 2(sqrt14)/14 = (2sqrt14)/14 = sqrt14/7
Combining all the terms again, we get:
(70 - 2(sqrt14) + sqrt14/7 + 2)/sqrt(14)
Now, we need to simplify the numerator:
70 - 2(sqrt14) + sqrt14/7 + 2 = 72 - sqrt14 + sqrt14/7
Simplifying further, we have:
72 - sqrt14 + sqrt14/7 = 72 + sqrt14(1 - 1/7) = 72 + sqrt14(6/7) = 72 + 6(sqrt14)/7
Finally, the expression becomes:
(72 + 6(sqrt14)/7)/sqrt14
We can rationalize the denominator by multiplying both the numerator and denominator by sqrt14:
[(72 + 6(sqrt14)/7) x (sqrt14)]/[sqrt14 x sqrt14] = (72(sqrt14) + 6(sqrt14)(sqrt14))/(sqrt14)^2
Simplifying further:
(72(sqrt14) + 6(sqrt14)(sqrt14))/(sqrt14)^2 = (72(sqrt14) + 6(14))/(sqrt14)^2 = (72(sqrt14) + 84)/(sqrt14)^2 = (72(sqrt14) + 84)/14
Finally, we can simplify the expression to:
72/14 + (84/14)(sqrt14) = 36/7 + 6(sqrt14)/7 = (36 + 6(sqrt14))/7
Therefore, the rationalized form of the expression is:
(36 + 6(sqrt14))/7