Questions A. Rationalise the denominators of the following expressions: $$ \begin{array}{ll} ext { 3. } \frac{2}{4+\sqrt{2}} & ext { 4. } \frac{6}{3-\sqrt{5}}\end{array} $$

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Questions A. Rationalise the denominators of the following expressions: $$ \begin{array}{ll}	ext { 3. } \frac{2}{4+\sqrt{2}} & 	ext { 4. } \frac{6}{3-\sqrt{5}}\end{array} $$

UpStudy AI · Accepted Answer

Sure, let's rationalize the denominators of the given expressions step by step.

### **3. Rationalize $$ \frac{2}{4 + \sqrt{2}} $$**

**Step 1:** Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $$4 + \sqrt{2}$$ is $$4 - \sqrt{2}$$.

$$
\frac{2}{4 + \sqrt{2}} \times \frac{4 - \sqrt{2}}{4 - \sqrt{2}}
$$

**Step 2:** Apply the multiplication.

$$
= \frac{2(4 - \sqrt{2})}{(4)^2 - (\sqrt{2})^2}
$$

**Step 3:** Simplify the denominator using the difference of squares formula $$a^2 - b^2$$.

$$
= \frac{8 - 2\sqrt{2}}{16 - 2} = \frac{8 - 2\sqrt{2}}{14}
$$

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

$$
= \frac{4 - \sqrt{2}}{7}
$$

**Final Answer:**

$$
\frac{2}{4 + \sqrt{2}} = \frac{4 - \sqrt{2}}{7}
$$

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### **4. Rationalize $$ \frac{6}{3 - \sqrt{5}} $$**

**Step 1:** Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $$3 - \sqrt{5}$$ is $$3 + \sqrt{5}$$.

$$
\frac{6}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}}
$$

**Step 2:** Apply the multiplication.

$$
= \frac{6(3 + \sqrt{5})}{(3)^2 - (\sqrt{5})^2}
$$

**Step 3:** Simplify the denominator using the difference of squares formula $$a^2 - b^2$$.

$$
= \frac{18 + 6\sqrt{5}}{9 - 5} = \frac{18 + 6\sqrt{5}}{4}
$$

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

$$
= \frac{9 + 3\sqrt{5}}{2}
$$

**Final Answer:**

$$
\frac{6}{3 - \sqrt{5}} = \frac{9 + 3\sqrt{5}}{2}
$$
$$ \frac{2}{4 + \sqrt{2}} = \frac{4 - \sqrt{2}}{7} $$ $$ \frac{6}{3 - \sqrt{5}} = \frac{9 + 3\sqrt{5}}{2} $$

Questions A. Rationalise the denominators of the following expressions: \( \begin{array}{ll}\text { 3. } \frac{2}{4+\sqrt{2}} & \text { 4. } \frac{6}{3-\sqrt{5}}\end{array} \)

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