how can you write the expression with rationalized denominator?
2+3sqrt3/3sqrt6
(2+3sqrt3)/3sqrt6 multiply by sqrt6/sqrt6
(2+3sqrt3)sqrt6/(3sqrt6*sqrt6)
(2+3sqrt3)sqrt6/18
would the answer be 2 3sqrt 6 + 9 3sqrt18 over 6
To rationalize the denominator of the expression (2 + 3√3) / (3√6), we can follow these steps:
Step 1: Determine the irrational numbers present in the denominator.
The denominator in this expression is 3√6. The √6 term is an irrational number because the square root of 6 cannot be expressed as a simple fraction.
Step 2: Multiply both the numerator and denominator by the conjugate of the irrational term.
The conjugate of the irrational term √6 is (-√6). Multiplying the numerator and denominator by (-√6) will eliminate the irrational term in the denominator.
(2 + 3√3) / (3√6) * (-√6) / (-√6)
Step 3: Simplify the expression.
Numerator:
(2 + 3√3) * (-√6) = -2√6 - 3√18
Simplifying the square root of 18:
√18 = √9 * √2 = 3√2
-2√6 - 3√18 = -2√6 - 3 * 3√2 = -2√6 - 9√2 = -11√2
Denominator:
(3√6) * (-√6) = -3√36
Simplifying the square root of 36:
√36 = √6 * √6 = 6
-3√36 = -3 * 6 = -18
Therefore, the rationalized expression is (-11√2) / (-18).
Note: If the expression were a fraction, it is typically recommended to simplify further by canceling out common factors. However, in this case, the expression is not a fraction, so we leave it in its rationalized form.