Questions A. Rationalise the denominators of the following expressions: $$ \begin{array}{ll} ext { 1. } \frac{3}{\sqrt{6}} & ext { 2. } \frac{2}{5 \sqrt{2}}\end{array} $$

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Questions A. Rationalise the denominators of the following expressions: $$ \begin{array}{ll}	ext { 1. } \frac{3}{\sqrt{6}} & 	ext { 2. } \frac{2}{5 \sqrt{2}}\end{array} $$

UpStudy AI · Accepted Answer

To **rationalize the denominator** of a fraction means to eliminate any irrational numbers (like square roots) from the denominator. This is typically done by multiplying both the numerator and the denominator by a suitable form of 1 that will remove the radical from the denominator.

Let's rationalize the denominators for each of the given expressions step-by-step.

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### 1. Rationalize $$ \frac{3}{\sqrt{6}} $$

**Step 1:** Identify the irrational denominator, which is $$ \sqrt{6} $$.

**Step 2:** Multiply both the numerator and the denominator by $$ \sqrt{6} $$ to eliminate the square root in the denominator.

$$
\frac{3}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{3 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}}
$$

**Step 3:** Simplify the denominator. Recall that $$ \sqrt{a} \times \sqrt{a} = a $$.

$$
\frac{3 \times \sqrt{6}}{6} = \frac{3\sqrt{6}}{6}
$$

**Step 4:** Simplify the fraction by dividing both the numerator and the denominator by 3.

$$
\frac{3\sqrt{6}}{6} = \frac{\sqrt{6}}{2}
$$

**Final Answer:**

$$
\frac{3}{\sqrt{6}} = \frac{\sqrt{6}}{2}
$$

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### 2. Rationalize $$ \frac{2}{5\sqrt{2}} $$

**Step 1:** Identify the irrational part in the denominator, which is $$ \sqrt{2} $$.

**Step 2:** Multiply both the numerator and the denominator by $$ \sqrt{2} $$ to eliminate the square root in the denominator.

$$
\frac{2}{5\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2 \times \sqrt{2}}{5 \times \sqrt{2} \times \sqrt{2}}
$$

**Step 3:** Simplify the denominator. Recall that $$ \sqrt{2} \times \sqrt{2} = 2 $$.

$$
\frac{2\sqrt{2}}{5 \times 2} = \frac{2\sqrt{2}}{10}
$$

**Step 4:** Simplify the fraction by dividing both the numerator and the denominator by 2.

$$
\frac{2\sqrt{2}}{10} = \frac{\sqrt{2}}{5}
$$

**Final Answer:**

$$
\frac{2}{5\sqrt{2}} = \frac{\sqrt{2}}{5}
$$

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By following these steps, you can rationalize denominators that contain square roots, making the expressions neater and often easier to work with in further calculations.
To rationalize the denominators: 1. $$ \frac{3}{\sqrt{6}} = \frac{\sqrt{6}}{2} $$ 2. $$ \frac{2}{5\sqrt{2}} = \frac{\sqrt{2}}{5} $$

Questions A. Rationalise the denominators of the following expressions: \( \begin{array}{ll}\text { 1. } \frac{3}{\sqrt{6}} & \text { 2. } \frac{2}{5 \sqrt{2}}\end{array} \)

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