simplify each expression. rationalize all denominators. assume that all variables are positive
(1-sqrt(3x))/(sqrt(6x))
Please show work!!!!
(1 - √(3x)/√(6x)
= (1 - √(3x)/√(6x) * √(6x)/√(6x)
= (√(6x) - (√18)x)/(6x)
= ( √(6x) - 3√2 x)/(6x)
To simplify and rationalize the denominator of the expression (1-sqrt(3x))/(sqrt(6x)), we can follow these steps:
Step 1: Multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the square root from the denominator.
The conjugate of sqrt(6x) is (-sqrt(6x)). So, we multiply both the numerator and the denominator by (-sqrt(6x)):
((1-sqrt(3x))/(sqrt(6x))) * ((-sqrt(6x))/(-sqrt(6x)))
This gives us:
((-sqrt(6x) + sqrt(18x^2))/(6x))
Step 2: Simplify the numerator:
The numerator now becomes (-sqrt(6x) + sqrt(18x^2)). We can further simplify this expression by factoring out sqrt(2x):
(-sqrt(6x) + sqrt(18x^2)) = -sqrt(6x) +sqrt(2x)*(sqrt(9x))
Using the property of square roots (sqrt(a) * sqrt(b) = sqrt(ab)), we have:
-sqrt(6x) + sqrt(2x)*(3x)
= -sqrt(6x) + 3sqrt(2x^2)
Step 3: Simplify the expression further:
The simplified expression with a rationalized denominator is:
(-sqrt(6x) + 3sqrt(2x^2))/(6x)