How can you write the expression with a rationalized denominator?
((3sqrt2)/(3sqrt6))
A. (3sqrt9)/3
B. ((2+(3sqrt9)))/6
C. (3sqrt9)/6
D. (3sqrt72)/3
((3sqrt2)/(3sqrt6)) can be written as ((3sqrt2)/(3sqrt2*sqrt3)). To rationalize the denominator, we can multiply both numerator and denominator by sqrt3, which gives:
((3sqrt2)/(3sqrt2*sqrt3)) * (sqrt3/sqrt3)
Simplifying the numerator and denominator:
= (3sqrt6)/(3*sqrt(2*3))
= (3sqrt6)/(3sqrt6)
= 1
Therefore, the expression with a rationalized denominator is 1, which is not one of the answer choices listed.
To rationalize the denominator of the expression ((3√2)/(3√6)), follow these steps:
Step 1: Simplify the radicals in the denominator.
√6 = √(2 * 3) = √2 * √3 = √2√3 = √(2∙3) = √(6)
Step 2: Rewrite the expression with the simplified denominator.
((3√2)/(3√6)) = ((3√2)/(3√(2∙3)))
Step 3: Multiply both the numerator and denominator by the conjugate of the denominator.
The conjugate of √(2∙3) is (√2∙√3).
((3√2)/(3√(2∙3))) * ((√2∙√3)/(√2∙√3)) = (3√2∙√2∙√3)/(3√(2∙3)∙√2∙√3)
Step 4: Simplify the expression further.
√2∙√2 = √(2∙2) = √4 = 2
√3∙√3 = √(3∙3) = √9 = 3
(3√2∙√2∙√3)/(3√(2∙3)∙√2∙√3) = (3∙2∙3)/(3∙√(2∙3)∙2∙√3)
Step 5: Simplify the expression and cancel out common factors.
(3∙2∙3)/(3∙√(2∙3)∙2∙√3) = (6∙3)/(3∙√(2∙3)∙2∙√3) = 18/(3∙2∙√(2∙3)∙√3)
(18)/(3∙2∙√(2∙3)∙√3) = 18/(6∙√(2∙3)∙√3)
Step 6: Multiply the factors in the denominator.
6∙√(2∙3)∙√3 = 6∙√(6)∙√3 = 6∙√(6∙3) = 6∙√(18)
Step 7: Rewrite the expression with the rationalized denominator.
18/(6∙√(2∙3)∙√3) = 18/(6∙√(18))
Step 8: Simplify the expression further.
6∙√(18) = 6∙√(9∙2) = 6∙√(9)∙√(2) = 6∙3∙√(2) = 18√(2)
18/(6∙√(18)) = 18/(18√(2))
Step 9: Cancel out the common factors.
18/(18√(2)) = 1/√(2)
Step 10: Rationalize the denominator by multiplying the numerator and denominator by √2.
(1/√(2)) * (√2/√2) = (√2)/(√(2*2)) = (√2)/(2)
Therefore, the expression ((3√2)/(3√6)) can be written as (√2)/(2), which is answer choice B.