Simplify by rationalizing the denominator: 1. $$ \frac{3-3 \sqrt{3 a}}{4 \sqrt{8 a}} $$

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UpStudy AI · Accepted Answer

We start with
$$
\frac{3-3 \sqrt{3a}}{4 \sqrt{8a}}.
$$
 
**Step 1. Factor the numerator and simplify the denominator**

Factor out $$3$$ in the numerator:
$$
\frac{3(1-\sqrt{3a})}{4 \sqrt{8a}}.
$$
Notice that the square root in the denominator can be simplified:
$$
\sqrt{8a} = \sqrt{8}\sqrt{a} = 2\sqrt{2}\sqrt{a}.
$$
Thus, the expression becomes:
$$
\frac{3(1-\sqrt{3a})}{4\cdot 2\sqrt{2a}} = \frac{3(1-\sqrt{3a})}{8\sqrt{2a}}.
$$
 
**Step 2. Rationalize the denominator**

We eliminate the square root in the denominator by multiplying the numerator and the denominator by $$\sqrt{2a}$$:
$$
\frac{3(1-\sqrt{3a})}{8\sqrt{2a}} \times \frac{\sqrt{2a}}{\sqrt{2a}}.
$$
Multiplying the numerator:
$$
3(1-\sqrt{3a})\sqrt{2a},
$$
and the denominator becomes:
$$
8\sqrt{2a}\cdot\sqrt{2a} = 8(2a)=16a.
$$
Thus, the expression simplifies to:
$$
\frac{3\sqrt{2a}(1-\sqrt{3a})}{16a}.
$$
 
This is the simplified form with a rationalized denominator.
$$ \frac{3\sqrt{2a}(1-\sqrt{3a})}{16a} $$

Simplify by rationalizing the denominator: 1. \( \frac{3-3 \sqrt{3 a}}{4 \sqrt{8 a}} \)

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