QuestionRationalize: $\quad \frac{11}{3 \sqrt{12}}$

Studdy Solution

STEP 1

1. The goal is to rationalize the denominator of the fraction.
2. Rationalizing involves removing the square root from the denominator.

STEP 2

1. Identify the form of the denominator that needs rationalization.
2. Multiply the numerator and the denominator by a term that will eliminate the square root in the denominator.
3. Simplify the resulting expression.

STEP 3

Identify the form of the denominator. The denominator is $3 \sqrt{12}$ .

STEP 4

To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt{12}$ :
$\frac{11}{3 \sqrt{12}} \times \frac{\sqrt{12}}{\sqrt{12}} = \frac{11 \sqrt{12}}{3 \times 12}$

STEP 5

Simplify the expression:
- Calculate the denominator: $3 \times 12 = 36$ . - The numerator is $11 \sqrt{12}$ .
So, the expression becomes:
$\frac{11 \sqrt{12}}{36}$

STEP 6

Further simplify $\sqrt{12}$ :
- $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$ .
Substitute back into the expression:
$\frac{11 \times 2 \sqrt{3}}{36} = \frac{22 \sqrt{3}}{36}$

STEP 7

Simplify the fraction:
- Divide both the numerator and the denominator by their greatest common divisor, which is 2:
$\frac{22 \sqrt{3}}{36} = \frac{11 \sqrt{3}}{18}$
The rationalized expression is:
$\frac{11 \sqrt{3}}{18}$

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