Simplify th radical expression by rationalizing the denominator:
2(square root)108 / (square root) 180y
Please show me how to set up and how to solve.
To rationalize the denominator of the given expression, we need to eliminate any square roots in the denominator.
Step 1: Simplify the radicals in the numerator and denominator.
Since 108 can be expressed as 36 x 3, we can simplify the numerator as follows:
2(sqrt(108)) = 2(sqrt(36 x 3)) = 2(sqrt(36) x sqrt(3)) = 2 x 6(sqrt(3)) = 12(sqrt(3))
Similarly, we can simplify the denominator:
(sqrt(180y)) = sqrt(36 x 5y) = sqrt(36) x sqrt(5y) = 6(sqrt(5y))
Now the expression becomes 12(sqrt(3)) / 6(sqrt(5y)).
Step 2: Divide both the numerator and denominator by the greatest common factor (GCF) to simplify further.
The GCF of 12 and 6 is 6, so dividing both by 6 gives:
12/6 = 2 and 6/6 = 1.
Therefore, the simplified expression is:
2(sqrt(3)) / (sqrt(5y))
To simplify the given radical expression by rationalizing the denominator, we need to eliminate the square root from the denominator. Here's how you can do it:
Step 1: Prime factorize the numbers inside the square roots:
√108 = √(2^2 * 3^3) = 2√3^2 * √3 = 2 * 3√3 = 6√3
√180y = √(2^2 * 3^2 * 5 * y) = 2 * 3 * √(5y) = 6√(5y)
Step 2: Substitute the simplified values back into the expression:
2√108 / √180y = 2(6√3) / (6√(5y))
Step 3: Cancel out the common factors between the numerator and denominator:
= 2(6√3) / (6√(5y))
= 12√3 / 6√(5y)
Step 4: Simplify the expression further:
= 2√3 / √(5y)
Thus, the simplified form of the given expression after rationalizing the denominator is 2√3 / √(5y).