Question
Questions A. Rationalise the denominators of the following expressions, without the use of a catculator: \( \begin{array}{ll}\text { 3. } \frac{2}{4+\sqrt{2}} & \text { 4. } \frac{6}{3-\sqrt{5}}\end{array} \)
Ask by Schwartz Pena. in South Africa
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
To rationalize the denominators:
1. For \( \frac{2}{4 + \sqrt{2}} \), multiply numerator and denominator by \( 4 - \sqrt{2} \):
\[
\frac{2(4 - \sqrt{2})}{(4 + \sqrt{2})(4 - \sqrt{2})} = \frac{8 - 2\sqrt{2}}{14} = \frac{4 - \sqrt{2}}{7}
\]
2. For \( \frac{6}{3 - \sqrt{5}} \), multiply numerator and denominator by \( 3 + \sqrt{5} \):
\[
\frac{6(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} = \frac{18 + 6\sqrt{5}}{4} = \frac{9 + 3\sqrt{5}}{2}
\]
Final answers:
\[
\frac{4 - \sqrt{2}}{7} \quad \text{and} \quad \frac{9 + 3\sqrt{5}}{2}
\]
Solution
To rationalize the denominators of the given expressions without using a calculator, we'll use the method of multiplying by the **conjugate** of the denominator. The conjugate of a binomial \( a + b \) is \( a - b \), and vice versa. This method helps eliminate the square roots in the denominator.
Let's tackle each expression step-by-step.
---
### **3. \( \frac{2}{4 + \sqrt{2}} \)**
**Step 1: Identify the Conjugate**
The denominator is \( 4 + \sqrt{2} \). Its conjugate is \( 4 - \sqrt{2} \).
**Step 2: Multiply Numerator and Denominator by the Conjugate**
\[
\frac{2}{4 + \sqrt{2}} \times \frac{4 - \sqrt{2}}{4 - \sqrt{2}} = \frac{2(4 - \sqrt{2})}{(4 + \sqrt{2})(4 - \sqrt{2})}
\]
**Step 3: Simplify the Denominator**
\[
(4 + \sqrt{2})(4 - \sqrt{2}) = 4^2 - (\sqrt{2})^2 = 16 - 2 = 14
\]
**Step 4: Expand the Numerator**
\[
2(4 - \sqrt{2}) = 8 - 2\sqrt{2}
\]
**Step 5: Combine the Results**
\[
\frac{8 - 2\sqrt{2}}{14} = \frac{8}{14} - \frac{2\sqrt{2}}{14} = \frac{4}{7} - \frac{\sqrt{2}}{7} = \frac{4 - \sqrt{2}}{7}
\]
**Final Answer:**
\[
\boxed{\dfrac{4 - \sqrt{2}}{7}}
\]
---
### **4. \( \frac{6}{3 - \sqrt{5}} \)**
**Step 1: Identify the Conjugate**
The denominator is \( 3 - \sqrt{5} \). Its conjugate is \( 3 + \sqrt{5} \).
**Step 2: Multiply Numerator and Denominator by the Conjugate**
\[
\frac{6}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}} = \frac{6(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}
\]
**Step 3: Simplify the Denominator**
\[
(3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4
\]
**Step 4: Expand the Numerator**
\[
6(3 + \sqrt{5}) = 18 + 6\sqrt{5}
\]
**Step 5: Combine the Results**
\[
\frac{18 + 6\sqrt{5}}{4} = \frac{18}{4} + \frac{6\sqrt{5}}{4} = \frac{9}{2} + \frac{3\sqrt{5}}{2} = \frac{9 + 3\sqrt{5}}{2}
\]
**Final Answer:**
\[
\boxed{\dfrac{9 + 3\sqrt{5}}{2}}
\]
---
These rationalized forms eliminate the radicals from the denominators, making the expressions simpler and more standardized.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To rationalize the denominator of \( \frac{2}{4+\sqrt{2}} \), multiply the numerator and denominator by the conjugate of the denominator, which is \( 4-\sqrt{2} \). Hence, \[ \frac{2}{4+\sqrt{2}} \times \frac{4-\sqrt{2}}{4-\sqrt{2}} = \frac{2(4-\sqrt{2})}{(4+\sqrt{2})(4-\sqrt{2})} = \frac{8-2\sqrt{2}}{16-2} = \frac{8-2\sqrt{2}}{14} = \frac{4-\sqrt{2}}{7}. \] For the expression \( \frac{6}{3-\sqrt{5}} \), similarly, multiply by the conjugate: \[ \frac{6}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} = \frac{6(3+\sqrt{5})}{(3-\sqrt{5})(3+\sqrt{5})} = \frac{18 + 6\sqrt{5}}{9-5} = \frac{18 + 6\sqrt{5}}{4} = \frac{9 + 3\sqrt{5}}{2}. \]
