Rational and Irrational Numbers - Q&A

EXERCISE 1 (A)

1. Insert two rational numbers between :
(i) ³/₈ and ⁷/₁₂

Answer:
To find rational numbers between fractions, we first make their denominators equal.
LCM of 8 and 12 is 24.

³/₈ = ^{(3 × 3)}/_{(8 × 3)} = ⁹/₂₄

⁷/₁₂ = ^{(7 × 2)}/_{(12 × 2)} = ¹⁴/₂₄

We need to insert two rational numbers between ⁹/₂₄ and ¹⁴/₂₄.

The integers between 9 and 14 are 10, 11, 12, 13.

So, two rational numbers can be ¹⁰/₂₄ and ¹¹/₂₄.

Simplifying ¹⁰/₂₄ gives ⁵/₁₂.

Two rational numbers are: ⁵/₁₂ and ¹¹/₂₄.

(ii) ¹/₃ and ¹/₄

Answer:
Make denominators equal. LCM of 3 and 4 is 12.

¹/₃ = ⁴/₁₂

¹/₄ = ³/₁₂

Since there are no integers between 3 and 4, multiply numerator and denominator by (n+1) where n=2. So, multiply by 3.

⁴/₁₂ = ^{(4 × 3)}/_{(12 × 3)} = ¹²/₃₆

³/₁₂ = ^{(3 × 3)}/_{(12 × 3)} = ⁹/₃₆

Rational numbers between ⁹/₃₆ and ¹²/₃₆ are ¹⁰/₃₆ and ¹¹/₃₆.

Simplifying ¹⁰/₃₆ gives ⁵/₁₈.

Two rational numbers are: ⁵/₁₈ and ¹¹/₃₆.

2. Insert three rational numbers between :

(i) ²/₅ and ³/₇

Answer:

LCM of 5 and 7 is 35.

²/₅ = ¹⁴/₃₅

³/₇ = ¹⁵/₃₅

To find 3 numbers, multiply numerators and denominators by 4 (3+1).

¹⁴/₃₅ = ⁵⁶/₁₄₀

¹⁵/₃₅ = ⁶⁰/₁₄₀

Numbers between 56 and 60 are 57, 58, 59.

Three rational numbers are: ⁵⁷/₁₄₀, ⁵⁸/₁₄₀ (or ²⁹/₇₀), and ⁵⁹/₁₄₀.

(ii) ⁴/₁₁ and ⁹/₁₆

Answer:

LCM of 11 and 16 is 176.

⁴/₁₁ = ⁶⁴/₁₇₆

⁹/₁₆ = ⁹⁹/₁₇₆

Three numbers between them: ⁶⁵/₁₇₆, ⁶⁶/₁₇₆ (³/₈), ⁶⁷/₁₇₆.

3. (i) Find three rational numbers between 5 and -2.

Answer:
The integers between -2 and 5 include -1, 0, 1, 2, 3, 4.
We can pick any three.
Three rational numbers are: 0, 1, 2.

(ii) Find three rational numbers between -³/₄ and ¹/₂ .

Answer:
LCM of 4 and 2 is 4.

-³/₄ = -³/₄

¹/₂ = ²/₄

Integers between -3 and 2 are -2, -1, 0, 1.

We can write the rational numbers as: -²/₄, -¹/₄, ⁰/₄.

Three rational numbers are: -¹/₂, -¹/₄, and 0.

4. Insert 4 rational numbers between 5 and 8.

Answer:
We can use the formula d = (y - x) / (n + 1).

Here x = 5, y = 8, n = 4.

d = (8 - 5) / 5 = ³/₅ = 0.6

The numbers are:
x + d = 5 + 0.6 = 5.6 (or ²⁸/₅)

x + 2d = 5 + 1.2 = 6.2 (or ³¹/₅)

x + 3d = 5 + 1.8 = 6.8 (or ³⁴/₅)

x + 4d = 5 + 2.4 = 7.4 (or ³⁷/₅)

The four rational numbers are: 5.6, 6.2, 6.8, 7.4.

5. Insert 5 rational numbers between ¹/₃ and ⁵/₉

Answer:
Convert to like fractions.

¹/₃ = ³/₉

We have ³/₉ and ⁵/₉

To find 5 numbers, multiply numerators and denominators by 3 (to create enough gap).

³/₉ = ⁹/₂₇

⁵/₉ = ¹⁵/₂₇

Numbers between 9 and 15 are 10, 11, 12, 13, 14.

The five rational numbers are: ¹⁰/₂₇, ¹¹/₂₇, ¹²/₂₇ (or ⁴/₉), ¹³/₂₇, ¹⁴/₂₇

6. Insert 6 rational numbers between 4.6 and 8.4.

Answer:
We can simply pick terminating decimals between 4.6 and 8.4.
Let's choose: 5, 5.5, 6, 6.5, 7, 7.5.
All these lie between 4.6 and 8.4.

7. Insert 7 rational numbers between 1 and 2.

Answer:
x = 1, y = 2, n = 7.

d = (2 - 1) / (7 + 1) = ¹/₈

The numbers are 1 + ¹/₈, 1 + ²/₈, ..., 1 + ⁷/₈

The rational numbers are: ⁹/₈, ¹⁰/₈ (⁵/₄), ¹¹/₈, ¹²/₈ (³/₂), ¹³/₈, ¹⁴/₈ (⁷/₄), ¹⁵/₈

8. Insert 8 rational numbers between 1.8 and 3.6.

Answer:
We can use a step of 0.2.
1.8 + 0.2 = 2.0
2.0 + 0.2 = 2.2
... and so on.
Eight rational numbers are: 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4.

9. Arrange ⁵/₇, ^-12/₁₇, ⁹/₁₁ and ^-2/₃ in the ascending order of their magnitudes. Also, find the difference between the largest and the smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.

Answer:
Convert to decimals to compare magnitudes:

⁵/₇ ≈ 0.714

^-12/₁₇ ≈ -0.706

⁹/₁₁ ≈ 0.818

^-2/₃ ≈ -0.666

Ascending order (smallest to largest): -0.706 < -0.666 < 0.714 < 0.818

Order: ^-12/₁₇, ^-2/₃, ⁵/₇, ⁹/₁₁.

Largest number = ⁹/₁₁

Smallest number = ^-12/₁₇

Difference = ⁹/₁₁ - (^-12/₁₇) = ⁹/₁₁ + ¹²/₁₇

LCM = 187

= ^{(9 × 17 + 12 × 11)}/₁₈₇ = ^{(153 + 132)}/₁₈₇ = ²⁸⁵/₁₈₇

In decimal: 285 ÷ 187 ≈ 1.524...
Correct to one decimal place: 1.5.

10. Arrange ⁵/₈, ^-3/₁₆, ^-1/₄ and ¹⁷/₃₂ in the descending order of their magnitudes. Also, find the sum of the lowest and the largest of these rational numbers. Express the result obtained as a decimal fraction correct to two decimal places.

Answer:
Make denominators 32

⁵/₈ = ²⁰/₃₂

^-3/₁₆ = ^-6/₃₂

^-1/₄ = ^-8/₃₂

¹⁷/₃₂ = ¹⁷/₃₂

Comparing numerators: 20, 17, -6, -8.
Descending order (largest to smallest): ⁵/₈, ¹⁷/₃₂, ^-3/₁₆, ^-1/₄

Largest = ⁵/₈ (or ²⁰/₃₂)

Lowest = ^-1/₄ (or ^-8/₃₂)

Sum = ²⁰/₃₂ + (^-8/₃₂) = ¹²/₃₂ = ³/₈

Decimal: 3 ÷ 8 = 0.375
Correct to two decimal places: 0.38.

EXERCISE 1 (B)

1. State, which of the following decimal numbers are pure recurring decimals and which are mixed recurring decimals :
(i) 0.083 (bar on 083)

Answer: Pure recurring decimal (all digits after decimal repeat).

(ii) 0.083 (bar on 83)

Answer: Mixed recurring decimal (0 does not repeat).

(iii) 0.227 (bar on 227)

Answer: Pure recurring decimal.

(iv) 3.54 (bar on 4)

Answer: Mixed recurring decimal.

(v) 2.81 (bar on 81)

Answer: Pure recurring decimal.

2. Represent as a decimal number :

(i) ²/₇

Answer:
Dividing 2 by 7:
20 ÷ 7 = 2 rem 6
60 ÷ 7 = 8 rem 4
40 ÷ 7 = 5 rem 5
50 ÷ 7 = 7 rem 1
10 ÷ 7 = 1 rem 3
30 ÷ 7 = 4 rem 2 (Repeats)
Result: 0.285714 (bar on 285714)

(ii) ⁵/₂₄

Answer:
5 ÷ 24 = 0.208333... = 0.2083 (bar on 3)

(iii) ⁴/₉

Answer:
4 ÷ 9 = 0.444... = 0.4 (bar on 4)

(iv) ⁸/₁₃

Answer:
8 ÷ 13 = 0.615384 (bar on 615384)

3. Express each of the following as a rational number i.e. in the form ^a/_b where a, b ∈ Z and b ≠ 0.
(i) 0.53 (bar on 3)

Answer:
Let x = 0.5333...
10x = 5.333...
100x = 53.333...

90x = 48 ⇒ x = ⁴⁸/₉₀ = ⁸/₁₅

(ii) 0.227 (bar on 27)

Answer:
Let x = 0.22727...
10x = 2.2727...
1000x = 227.2727...

990x = 225 ⇒ x = ²²⁵/₉₉₀ = ⁵/₂₂

(iii) 0.2104 (bar on 104)

Answer:
Let x = 0.2104104...
10x = 2.104104...
10000x = 2104.104...

9990x = 2102 ⇒ x = ²¹⁰²/₉₉₉₀ = ¹⁰⁵¹/₄₉₉₅

(iv) 3.52 (bar on 2)

Answer:
Let x = 3.5222...
10x = 35.222...
100x = 352.222...

90x = 317 ⇒ x = ³¹⁷/₉₀ (or 3⁴⁷/₉₀)

(v) 2.24689 (bar on 689)

Answer:
Let x = 2.24689689...
100x = 224.689689...
100000x = 224689.689...

99900x = 224465 ⇒ x = ²²⁴⁴⁶⁵/₉₉₉₀₀ = 2 ⁴⁹³³/₁₉₉₈₀

(vi) 0.572 (bar on 572)

Answer:
Let x = 0.572572...
1000x = 572.572...

999x = 572 ⇒ x = ⁵⁷²/₉₉₉

(vii) 0.158 (bar on 58)

Answer:
Let x = 0.15858...
10x = 1.5858...
1000x = 158.5858...

990x = 157 ⇒ x = ¹⁵⁷/₉₉₀

(viii) 0.0384 (bar on 84)

Answer:
Let x = 0.038484...
100x = 3.8484...
10000x = 384.8484...

9900x = 381 ⇒ x = ³⁸¹/₉₉₀₀ = ¹²⁷/₃₃₀₀

4. Find the decimal representation of ¹/₇ and ²/₇ . Deduce from the decimal representation of ¹/₇ , without actual calculation, the decimal representation of ³/₇ , ⁴/₇ , ⁵/₇ and ⁶/₇ .

Answer:

¹/₇ = 0.142857 (bar on 142857)

²/₇ = 2 × ¹/₇ = 0.285714 (bar on 285714)

Deductions:
The cyclic order of digits is 1-4-2-8-5-7.

³/₇ starts with next higher digit after 2, which is 4: 0.428571 (bar on 428571)

⁴/₇ starts with next higher digit after 4, which is 5: 0.571428 (bar on 571428)

⁵/₇ starts with next higher digit after 5, which is 7: 0.714285 (bar on 714285)

⁶/₇ starts with next higher digit after 7, which is 8: 0.857142 (bar on 857142)

5. Without doing any actual division, find which of the following rational numbers have terminating decimal representation:

(i) ⁷/₁₆

Answer: Denominator 16 = 2⁴. Only factors of 2. Terminating.

(ii) ²³/₁₂₅

Answer: Denominator 125 = 5³. Only factors of 5. Terminating.

(iii) ⁹/₁₄

Answer: Denominator 14 = 2 × 7. Contains factor 7. Non-terminating.

(iv) ³²/₄₅

Answer: Denominator 45 = 9 × 5 = 3² × 5. Contains factor 3. Non-terminating.

(v) ⁴³/₅₀

Answer: Denominator 50 = 2 × 5². Only factors of 2 and 5. Terminating.

(vi) ¹⁷/₄₀

Answer: Denominator 40 = 2³ × 5. Only factors of 2 and 5. Terminating.

(vii) ⁶¹/₇₅

Answer: Denominator 75 = 3 × 5². Contains factor 3. Non-terminating.

(viii) ¹²³/₂₅₀

Answer: Denominator 250 = 2 × 5³. Only factors of 2 and 5. Terminating.

EXERCISE 1 (C)

1. State, whether the following numbers are rational or not:

(i) (2 + √2)²

Answer:

= 2² + (√2)² + 2(2)(√2)

= 4 + 2 + 4√2 = 6 + 4√2

Since √2 is irrational, the sum is Irrational.

(ii) (3 - √3)²

Answer:

= 9 + 3 - 6√3 = 12 - 6√3

Irrational.

(iii) (5 + √5)(5 - √5)

Answer:

= 5² - (√5)² = 25 - 5 = 20

Rational.

(iv) (√3 - √2)²

Answer:

= 3 + 2 - 2√6 = 5 - 2√6

Irrational.

(v) (³/_2√2)²

Answer:

= ⁹/_{(4 × 2)} = ⁹/₈

Rational.

(vi) (^√7/_6√2)²

Answer:

= ⁷/_{(36 × 2)} = ⁷/₇₂

Rational.

2. Find the square of:

(i) ^3√5/₅

Answer:

Square = (^3√5/₅)² = ^{(9 × 5)}/₂₅ = ⁴⁵/₂₅ = ⁹/₅ (or 1.8)

(ii) √3 + √2

Answer:

Square = (√3 + √2)² = 3 + 2 + 2√6 = 5 + 2√6.

(iii) √5 - 2

Answer:

Square = (√5 - 2)² = 5 + 4 - 4√5 = 9 - 4√5.

(iv) 3 + 2√5

Answer:

Square = (3 + 2√5)² = 9 + (4 × 5) + 2(3)(2√5) = 9 + 20 + 12√5 = 29 + 12√5.

3. State, in each case, whether true or false:

(i) √2 + √3 = √5

Answer: False (Sum of square roots is not square root of sum).

(ii) 2√4 + 2 = 6

Answer: 2(2) + 2 = 4 + 2 = 6. True.

(iii) 3√7 - 2√7 = √7

Answer: (3-2)√7 = 1√7. True.

(iv) ²/₇ is an irrational number.

Answer: False (It is a ratio of integers).

(v) ⁵/₁₁ is a rational number.

Answer: True.

(vi) All rational numbers are real numbers.

Answer: True.

(vii) All real numbers are rational numbers.

Answer: False (Some are irrational).

(viii) Some real numbers are rational numbers.

Answer: True.

4. Given universal set = {-6, -5³/₄, -√4, -³/₅, -³/₈, 0, ⁴/₅, 1, 1²/₃, √8, 3.01, π, 8.47} . From the given set, find:
(i) set of rational numbers

Answer:
Rational numbers are those that can be expressed as a/b or terminating/recurring decimals.
-√4 = -2 (Rational).
√8 is Irrational.
π is Irrational.

Set = {-6, -5³/₄, -√4, -³/₅, -³/₈, 0, ⁴/₅, 1, 1²/₃, 3.01, 8.47}

(ii) set of irrational numbers

Answer: {√8, π}

(iii) set of integers

Answer: {-6, -√4, 0, 1} (Note: -√4 is -2).

(iv) set of non-negative integers

Answer: {0, 1}

5. Use division method to show that √3 and √5 are irrational numbers.

Answer:
To show they are irrational, we compute the square root by long division and observe it is non-terminating and non-recurring.
For √3:
1.732...
The division continues indefinitely without a repeating pattern. Therefore, it is irrational.
For √5:
2.236...
Similarly, the digits do not terminate or repeat periodically. Therefore, it is irrational.

6. Use method of contradiction to show that √3 and √5 are irrational numbers.

Answer:
For √3:

Assume √3 is rational. Then √3 = a/b where a, b are coprime integers.

3 = a²/b² ⇒ a² = 3b²

So a² is divisible by 3, which implies a is divisible by 3. Let a = 3c.

(3c)² = 3b² ⇒ 9c² = 3b² ⇒ b² = 3c²

So b² is divisible by 3, implies b is divisible by 3.

Thus, a and b share a common factor 3, contradicting the assumption they are coprime. So √3 is irrational.

For √5:

Similar logic. Assume √5 = a/b. a² = 5b². a is divisible by 5. Let a = 5c.

25c² = 5b² ⇒ b² = 5c². b is divisible by 5.

Contradiction. So √5 is irrational.

7. Write a pair of irrational numbers whose sum is irrational.

Answer: √2 and √3 (Sum = √2 + √3, which is irrational).

8. Write a pair of irrational numbers whose sum is rational.

Answer: (2 + √3) and (2 - √3). Sum = 4.

9. Write a pair of irrational numbers whose difference is irrational.

Answer: √5 and √2. Difference = √5 - √2.

10. Write a pair of irrational numbers whose difference is rational.

Answer: (3 + √2) and √2. Difference = 3.

11. Write a pair of irrational numbers whose product is irrational.

Answer: √2 and √3. Product = √6.

12. Write a pair of irrational numbers whose product is rational.

Answer: √2 and √8. Product = √16 = 4.

13. Write in ascending order:
(i) 3√5 and 4√3

Answer:
Square both numbers to compare.
(3√5)² = 9 × 5 = 45.
(4√3)² = 16 × 3 = 48.
Since 45 < 48, 3√5 < 4√3.

(ii) 2 ³√5 and 3 ³√2

Answer:
Cube both numbers.
(2 ³√5)³ = 8 × 5 = 40.
(3 ³√2)³ = 27 × 2 = 54.
40 < 54, so 2 ³√5 < 3 ³√2.

(iii) 7√3 and 8√2

Answer:
(7√3)² = 49 × 3 = 147.
(8√2)² = 64 × 2 = 128.
128 < 147, so 8√2 < 7√3.

14. Write in descending order :
(i) 2 ⁴√6 and 3 ⁴√2

Answer:
Raise to power 4.
(2 ⁴√6)⁴ = 16 × 6 = 96.
(3 ⁴√2)⁴ = 81 × 2 = 162.
162 > 96, so 3 ⁴√2 > 2 ⁴√6.

(ii) 7√3 and 3√7

Answer:
(7√3)² = 49 × 3 = 147.
(3√7)² = 9 × 7 = 63.
147 > 63, so 7√3 > 3√7.

15. Compare:
(i) ⁶√15 and ⁴√12

Answer:
LCM of orders 6 and 4 is 12.
⁶√15 = 15^1/6 = 15^2/12 = (15²)^1/12 = ¹²√225.
⁴√12 = 12^1/4 = 12^3/12 = (12³)^1/12 = ¹²√1728.
Since 1728 > 225, ⁴√12 > ⁶√15.

(ii) √24 and ³√35

Answer:
LCM of 2 and 3 is 6.
√24 = 24^1/2 = 24^3/6 = (24³)^1/6 = √(13824).
³√35 = 35^1/3 = 35^2/6 = (35²)^1/6 = √(1225).
13824 > 1225, so √24 > ³√35.

16. Insert two irrational numbers between 5 and 6.

Answer:
5 = √25 and 6 = √36.
Any square root of a non-perfect square between 25 and 36 will work.
Examples: √26, √27.

17. Insert five irrational numbers between 2√5 and 3√3

Answer:
2√5 = √(4 × 5) = √20.
3√3 = √(9 × 3) = √27.
We need irrational numbers between √20 and √27.
We can choose: √21, √22, √23, √24, √26.

18. Write two rational numbers between √2 and √3.

Answer:
√2 ≈ 1.414.
√3 ≈ 1.732.
We can pick terminating decimals: 1.5, 1.6.
Rational numbers: 1.5 (³/₂) and 1.6 (⁸/₅).

19. Write three rational numbers between √3 and √5.

Answer:
√3 ≈ 1.732.
√5 ≈ 2.236.
Rational numbers can be 1.8, 1.9, 2.0.
Answer: 1.8, 1.9, 2.

EXERCISE 1 (D)

1. State, with reason, which of the following are surds and which are not :
(i) √180

Answer:
√180 = √(36 × 5) = 6√5. Since √5 is irrational, this is a Surd.

(ii) ⁴√27

Answer:
27 is not a perfect 4th power. 3³ is not a 4th power. Result is irrational. Surd.

(iii) ⁵√128

Answer:
128 = 2⁷. Not a perfect 5th power. Result is irrational. Surd.

(iv) ³√64

Answer:
64 = 4³. So ³√64 = 4. This is rational. Not a surd.

(v) ³√25 · ³√40

Answer:
= ³√(25 × 40) = ³√1000 = 10. Rational. Not a surd.

(vi) ³√-125

Answer:
-125 = (-5)³. Root is -5. Rational. Not a surd.

(vii) √π

Answer:
π is irrational, not rational. Definition of surd requires the radicand to be rational. Hence, Not a surd (though it is irrational).

(viii) √(3 + √2)

Answer:
Radicand (3 + √2) is irrational. Hence Not a surd (it is an irrational number, but technically not a surd by definition which requires positive rational radicand).

2. Write the lowest rationalising factor of:
(i) 5√2

Answer: √2

(ii) √24

Answer: √24 = 2√6. Factor is √6.

(iii) √5 - 3

Answer: (Order changed to standard surd form usually) or simply conjugate: √5 + 3. Note: (√5-3)(√5+3) = 5-9 = -4.

(iv) 7 - √7

Answer: 7 + √7.

(v) √18 - √50

Answer: Simplify first: 3√2 - 5√2 = -2√2. Factor is √2.

(vi) √5 - √2

Answer: √5 + √2.

(vii) √13 + 3

Answer: √13 - 3.

(viii) 15 - 3√2

Answer: 3(5 - √2). Factor for (5 - √2) is 5 + √2.

(ix) 3√2 + 2√3

Answer: 3√2 - 2√3.

3. Rationalise the denominators of:
(i) ³/_√5

Answer:

Multiply by √5/√5

= ^3√5/₅

(ii) ^2√3/_√5

Answer:

Multiply by √5/√5

= ^2√15/₅

(iii) ¹/_{(√3 - √2)}

Answer:

Multiply by (√3 + √2)

= (√3 + √2) / (3 - 2) = √3 + √2

(iv) ³/_{(√5 + √2)}

Answer:

Multiply by (√5 - √2)

= 3(√5 - √2) / (5 - 2) = 3(√5 - √2) / 3 = √5 - √2

(v) ^{(2 - √3)}/_{(2 + √3)}

Answer:

Multiply num and den by (2 - √3)

= (2 - √3)² / (4 - 3) = 4 + 3 - 4√3 = 7 - 4√3

(vi) ^{(√3 + 1)}/_{(√3 - 1)}

Answer:

Multiply by (√3 + 1)

= (√3 + 1)² / (3 - 1) = (3 + 1 + 2√3) / 2 = (4 + 2√3) / 2 = 2 + √3

(vii) ^{(√3 - √2)}/_{(√3 + √2)}

Answer:

Multiply by (√3 - √2)

= (√3 - √2)² / (3 - 2) = 3 + 2 - 2√6 = 5 - 2√6

(viii) ^{(√6 - √5)}/_{(√6 + √5)}

Answer:

Multiply by (√6 - √5)

= (√6 - √5)² / (6 - 5) = 6 + 5 - 2√30 = 11 - 2√30

(ix) ^{(2√5 + 3√2)}/_{(2√5 - 3√2)}

Answer:
Multiply by (2√5 + 3√2).
Numerator = (2√5 + 3√2)² = 20 + 18 + 12√10 = 38 + 12√10.
Denominator = (2√5)² - (3√2)² = 20 - 18 = 2.
Result = (38 + 12√10) / 2 = 19 + 6√10

4. Find the values of 'a' and 'b' in each of the following:

(i) ^{(2 + √3)}/_{(2 - √3)} = a + b√3

Answer:
LHS = (2 + √3)² / (4 - 3) = 4 + 3 + 4√3 = 7 + 4√3.
Comparing with a + b√3:
a = 7, b = 4

(ii) ^{(√7 - 2)}/_{(√7 + 2)} = a√7 + b

Answer:
LHS = (√7 - 2)² / (7 - 4) = (7 + 4 - 4√7) / 3 = (11 - 4√7) / 3

= ¹¹/₃ - ⁴/₃√7

Matching a√7 + b:

a = -⁴/₃, b = ¹¹/₃

(iii) ³/_{(√3 - √2)} = a√3 - b√2

Answer:

LHS = 3(√3 + √2) / (3 - 2) = 3√3 + 3√2

Matching a√3 - b√2:

a = 3, -b = 3 ⇒ b = -3.

a = 3, b = -3

(iv) ^{(5 + 3√2)}/_{(5 - 3√2)} = a + b√2

Answer:

LHS = (5 + 3√2)² / (25 - 18) = (25 + 18 + 30√2) / 7 = (43 + 30√2) / 7

= ⁴³/₇ + ³⁰/₇√2

a = ⁴³/₇, b = ³⁰/₇

5. Simplify:

(i) ²²/_{(2√3 + 1)} + ¹⁷/_{(2√3 - 1)}

Answer:

Term 1: 22(2√3 - 1) / (12 - 1) = 22(2√3 - 1) / 11 = 2(2√3 - 1) = 4√3 - 2

Term 2: 17(2√3 + 1) / (12 - 1) = 17(2√3 + 1) / 11 = ¹⁷/₁₁(2√3 + 1)

Sum = 4√3 - 2 + ³⁴/₁₁√3 + ¹⁷/₁₁

= (4 + ³⁴/₁₁)√3 + (-2 + ¹⁷/₁₁)

= ⁷⁸/₁₁√3 - ⁵/₁₁

(ii) ^√2/_{(√6 - √2)} - ^√3/_{(√6 + √2)}

Answer:
Term 1: √2(√6 + √2) / (6 - 2) = (√12 + 2) / 4 = (2√3 + 2) / 4 = ^{(√3 + 1)}/₂

Term 2: √3(√6 - √2) / (6 - 2) = (√18 - √6) / 4 = (3√2 - √6) / 4

Result: [2(√3 + 1) - (3√2 - √6)] / 4 = (2√3 + 2 - 3√2 + √6) / 4

6. If x = ^{(√5 - 2)}/_{(√5 + 2)} and y = ^{(√5 + 2)}/_{(√5 - 2).} ; find:

(i) x²

Answer:

x = (√5 - 2)² / (5 - 4) = 5 + 4 - 4√5 = 9 - 4√5

x² = (9 - 4√5)² = 81 + 80 - 72√5 = 161 - 72√5

(ii) y²

Answer:

y = (√5 + 2)² / 1 = 9 + 4√5

y² = (9 + 4√5)² = 81 + 80 + 72√5 = 161 + 72√5

(iii) xy

Answer:
Since y = 1/x, xy = 1.

(iv) x² + y² + xy

Answer:

(161 - 72√5) + (161 + 72√5) + 1

= 322 + 1 = 323.

7. If m = ¹/_{(3 - 2√2)} and n = ¹/_{(3 + 2√2)} , find:

(i) m²

Answer:

Rationalize m: m = (3 + 2√2) / (9 - 8) = 3 + 2√2

m² = (3 + 2√2)² = 9 + 8 + 12√2 = 17 + 12√2

(ii) n²

Answer:
Rationalize n: n = (3 - 2√2) / 1 = 3 - 2√2.
n² = (3 - 2√2)² = 9 + 8 - 12√2 = 17 - 12√2.

(iii) mn

Answer:
m × n = (3 + 2√2)(3 - 2√2) = 9 - 8 = 1.

8. If x = 2√3 + 2√2, find:
(i) ¹/_x

Answer:
¹/_x = 1 / (2√3 + 2√2) = (2√3 - 2√2) / (12 - 8) = 2(√3 - √2) / 4 = ^{(√3 - √2)}/₂.

(ii) x + ¹/_x

Answer:
x = 2(√3 + √2).
¹/_x = 0.5(√3 - √2).
Sum = 2√3 + 2√2 + 0.5√3 - 0.5√2 = 2.5√3 + 1.5√2 = ⁵/₂√3 + ³/₂√2.

(iii) (x + ¹/_x)²

Answer:
= (2.5√3 + 1.5√2)² = 6.25(3) + 2.25(2) + 2(2.5)(1.5)√6
= 18.75 + 4.5 + 7.5√6 = 23.25 + 7.5√6.

9. If x = 1 - √2, find the value of (x - ¹/_x)³.

Answer:

x = 1 - √2

¹/_x = 1 / (1 - √2) = (1 + √2) / (1 - 2) = -(1 + √2) = -1 - √2

x - ¹/_x = (1 - √2) - (-1 - √2) = 1 - √2 + 1 + √2 = 2

(x - ¹/_x)³ = 2³ = 8.

10. If x = 5 - 2√6, find: x² + ¹/_x²

Answer:

¹/_x = 1 / (5 - 2√6) = (5 + 2√6) / (25 - 24) = 5 + 2√6

x + ¹/_x = (5 - 2√6) + (5 + 2√6) = 10

x² + ¹/_x² = (x + ¹/_x)² - 2 = 10² - 2 = 100 - 2 = 98

11. Show that: ¹/_{(3 - 2√2)} - ¹/_{(2√2 - √7)} + ¹/_{(√7 - √6)} - ¹/_{(√6 - √5)} + ¹/_{(√5 - 2)} = 5.

Answer:
Rationalize each term:

1st: 3 + 2√2

2nd: (2√2 + √7) / (8 - 7) = 2√2 + √7

3rd: (√7 + √6) / (7 - 6) = √7 + √6

4th: (√6 + √5) / (6 - 5) = √6 + √5

5th: (√5 + 2) / (5 - 4) = √5 + 2

Substitute back (mind the signs):

(3 + 2√2) - (2√2 + √7) + (√7 + √6) - (√6 + √5) + (√5 + 2)
= 3 + 2√2 - 2√2 - √7 + √7 + √6 - √6 - √5 + √5 + 2
= 3 + 0 + 0 + 0 + 0 + 2
= 5.
Hence Proved.

12. Rationalise the denominator of: ¹/_{(√3 - √2 + 1)}

Answer:
Group terms: [(√3 + 1) - √2].
Multiply by [(√3 + 1) + √2].
Num: √3 + 1 + √2.
Denom: (√3 + 1)² - (√2)² = (3 + 1 + 2√3) - 2 = 4 + 2√3 - 2 = 2 + 2√3.
Expression: (1 + √2 + √3) / (2 + 2√3).
Rationalize again: Multiply by (2√3 - 2).
Num: (1 + √2 + √3)(2√3 - 2) = 2√3 - 2 + 2√6 - 2√2 + 6 - 2√3 = 4 - 2√2 + 2√6.
Denom: (2√3)² - 2² = 12 - 4 = 8.
Result: (4 - 2√2 + 2√6) / 8 = (2 - √2 + √6) / 4.

13. If √2 = 1.4 and √3 = 1.7, find the value of:
(i) ¹/_{(√3 - √2)}

Answer:
Rationalized = √3 + √2.
Value = 1.7 + 1.4 = 3.1.

(ii) ¹/_{(3 + 2√2)}

Answer:
Rationalized = 3 - 2√2.
Value = 3 - 2(1.4) = 3 - 2.8 = 0.2.