#### Simplify each of the following by rationalizing the denominator:

(i) (7 + 3√5) / (7 – 3√5)

(ii) (3 – 2√2) / (3 + 2√2)

(iii) (5 – 3√14) / (7 + 2√14)

**Solution:**

(i) (7 + 3√5) / (7 – 3√5)

Let us rationalize the denominator, we get

(7 + 3√5) / (7 – 3√5) = [(7 + 3√5) (7 + 3√5)] / [(7 – 3√5) (7 + 3√5)]

= [(7 + 3√5)^{2}] / [7^{2} – (3√5)^{2}]

= [7^{2} + (3√5)^{2} + 2.7. 3√5] / [49 – 9.5]

= [49 + 9.5 + 42√5] / [49 – 45]

= [49 + 45 + 42√5] / [4]

= [94 + 42√5] / 4

= 2[47 + 21√5]/4

= [47 + 21√5]/2

(ii) (3 – 2√2) / (3 + 2√2)

Let us rationalize the denominator, we get

(3 – 2√2) / (3 + 2√2) = [(3 – 2√2) (3 – 2√2)] / [(3 + 2√2) (3 – 2√2)]

= [(3 – 2√2)^{2}] / [3^{2} – (2√2)^{2}]

= [3^{2} + (2√2)^{2} – 2.3.2√2] / [9 – 4.2]

= [9 + 4.2 – 12√2] / [9 – 8]

= [9 + 8 – 12√2] / 1

= 17 – 12√2

(iii) (5 – 3√14) / (7 + 2√14)

Let us rationalize the denominator, we get

(5 – 3√14) / (7 + 2√14) = [(5 – 3√14) (7 – 2√14)] / [(7 + 2√14) (7 – 2√14)]

= [5(7 – 2√14) – 3√14 (7 – 2√14)] / [7^{2} – (2√14)^{2}]

= [35 – 10√14 – 21√14 + 6.14] / [49 – 4.14]

= [35 – 31√14 + 84] / [49 – 56]

= [119 – 31√14] / [-7]

= -[119 – 31√14] / 7

= [31√14 – 119] / 7

**More Solution:**

- Find the value of x2 + 5xy + y2.
- Choose the correct statement:
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- The product of any two irrational numbers is:
- Which of the following is an irrational number?
- The following is an irrational number?
- A rational number between √2 and √3 is:
- The decimal expansion of the rational number:
- 2√3 + √3 is equal to: