Rationalize the denominators when necessary. 4/squroot of x + 3?

Question

how do you do this problem? thanks

Answer 1

Is the denominator
sqrt (x + 3)
or
(sqrt x) + 3 ????
================
4 / sqrt (x + 3)

multiply the top and bottom by sqrt (x + 3)

4 sqrt (x + 3) / [sqrt (x + 3)]^2
4 sqrt (x + 3) / (x + 3)
====================
OR

4 / (sqrt x) + 3

multiply the top and bottom by (sqrt x) - 3

4 [(sqrt x) - 3] / [(sqrt x + 3)(sqrt x - 3)]
4 [(sqrt x) - 3] / (sqrt x)^2 - 3^2
4 [(sqrt x) - 3] / (x - 9)

Answer 2

The denominator is the part of the fraction below the line and the numerator is the part above the line.

Rationalizing the denominator means coming up with an expression with exactly the same value but without a square root in the denominator.

You need to understand two important principles.

Firstly, if you multiply both the numerator and the denominator by the same value, the value of the entire expression doesn't change. So, 3/4 = 0.75. (3*2)/(4*2) also equals 0.75. (I'm using * for multiply so as not to confuse it with the letter x.) Similarly, 3/x = (3*x)/(x*x) = (3*y)/(x*y) etc.

Secondly, when you multiply a square root by itself, you end up without the square root. So sqrt(3)*sqrt(3) = 3. That's the meaning of square root: what you multiply by itself to get the number. It also works with algebra sqrt(x)*sqrt(x) = x.

Now putting these together, if you multiply the numerator and the denominator by the square root expression, you will end up with an expression with the same value but without a square root in the denominator.

4 / sqrt(x+3)
= (4*sqrt(x+3)) / (sqrt(x+3)*sqrt(x+3))
= (4*sqrt(x+3)) / (x+3)

Answer 3

= 4sqrt(x+3)/(sqrt(x+3))^2
= 4sqrt(x+3)/(x+3)