Simplify the rational expression by rationalizing the denominator.
√12 / √8x
A. √3x / 4x
B. 3√4x / 4x
C. √6x / 2x
D. 2√3x / x
We can simplify the expression by rationalizing the denominator, which means to eliminate the radical in the denominator by multiplying both the numerator and denominator by a suitable expression. In this case, we can multiply by √2 on both the top and the bottom to obtain:
√12 / √8x = (√12 / √8x) * (√2 / √2)
= (√24 / 2√8x)
= (2√6 / 2 * 2√x)
= (√6 / 2√x)
Therefore, the simplified expression is:
C. √6x / 2x
To simplify the rational expression √12 / √8x, we need to rationalize the denominator.
First, let's simplify the numerator and denominator individually.
The square root of 12 can be further simplified as follows:
√12 = √(4 * 3) = √4 * √3 = 2√3.
Similarly, the square root of 8x can be further simplified as:
√8x = √(4 * 2x) = √4 * √(2x) = 2√(2x).
Now, we can rewrite the rational expression as:
(2√3) / (2√(2x)).
Next, we can cancel out the common factor of 2, giving us:
√3 / √(2x).
To rationalize the denominator, we can multiply both the numerator and the denominator by √(2x):
(√3 / √(2x)) * (√(2x) / √(2x)) = √(6x) / 2x.
Therefore, the simplified rational expression after rationalizing the denominator is √(6x) / 2x.
The correct answer is C. √6x / 2x.