Al-Khwarizmi invented Algebra and Surds (src: wikipedia) |

###
**Important points**

1. If a > 0, n and m are rational numbers, then

(i) a

^{m}. a^{n}= a^{m + n }
(ii) (a

^{m})^{n}= a^{mn}
(iii) a

^{m}/ a^{n}= a^{m + n }, m >n
(iv) a

^{m}b^{m}= (ab)^{m}
(v) a

^{0}= 1
2. If ‘a’ is a positive rational number and n is a positive integer such that,

**is rational number then**^{n}√a
(i)

**is called**^{n}√a*surd*or*radical*
(ii)

**√**is called*radical*or radical sign
(iii)

**n**is called*order*of radical
(iv)

(v)

**a**is called*radicand*.(v)

^{n}√a = a^{(1/n)}###
**Exercise 1.6**

**Q1: Find**

**(i) 64**

^{1/2}**(ii) 32**

^{1/5}**(iii) 125**

^{1/3}
Answer:

(i) 64

^{1/2}= (2^{6})^{1/2}= 2^{3}= 8
(ii) 32

^{1/5}= (2^{5})^{1/5}= 2
(iii) 125

^{1/3}= (5^{3})^{1/5}= 5**Q2: Find**

**(i) 9**

^{3/2}**(ii) 32**

^{2/5}**(iii) 16**

^{3/4}**(iv) 125**

^{-1/3}
Answer:

(i) 9

^{3/2}= (3^{2})^{3/2}= 3^{3}= 27
(ii) 32

^{2/5}= (2^{5})^{2/5}= 2^{2}=4
(iii) 16

(iv) 125

^{3/4}= (2^{4})^{3/4}= 2^{3}= 8(iv) 125

^{-1/3}= (5^{3})^{-1/3}= 5^{-1}= 1/5 = 0.20### Additional questions

**Q4: In the following equations determine whether a, b, c represent rational or irrational numbers.**

**(i) a**

^{3}= 27**(ii) b**

^{2}= 5**(iii) c**

^{2}= 0.09
Answer:

(i) a

^{3}= 27 ⇒ a^{3}= 3^{3}⇒ a = 3 ... (rational number)
(ii) b

^{2}= 5 ⇒ b = √5 ... (irrational number)
(iii) c

^{2}= 0.09 = 9/100 ⇒ c^{2}= √(3^{2}/10^{2}) = 3/10 = 0.3 ... (rational number)**Q 5: What is successive magnification in number line representation?**

Answer: The process of visualization of representation of numbers on the number line, through a magnifying glass is known as the process of successive magnification. The position of a real number with a terminating decimal expansion can be shown on the number line, by sufficient successive magnifications.

**Q6: Give two irrational numbers so that their**

(i) sum is not an irrational number.

(ii) difference is an irrational number.

(iii) product is an irrational number.

(iv) product is not an irrational number.

Answer:

(i) sum is not an irrational number.

e.g. sum of √5 and (-√5) is 0 which is not irrational.

(ii) difference is an irrational number.

e.g. √5 - √3, the difference is irrational.

(iii) product is an irrational number.

e.g. √3 x √5 = √(15) which is an irrational number.

(iv) product is not an irrational number.

e.g. √(12) x √3 = √(36) = 6 which is a rational number.

**Q7: Which one is greater √2 or**

^{3}√5 ?Answer: 2

^{1/2}and 5

^{1/3}

LCM of 2 and 3 is 6. Above surds can be written as:

2

^{3/6}and 5

^{2/6}

⇒ 8

^{1/6}and 25

^{1/6}(Now powers are same, we can compare radicands)

⇒ 25

^{1/6}> 8

^{1/6 }

or 5

^{1/3}> 2

^{1/2}

^{}

**Q8: If a**

^{x}= b, b^{y}= c and c^{z}= a then prove xyz = 1.Answer: Given, a

^{x}= b, b

^{y}= c and c

^{z}= a

⇒ (c

^{z})

^{x}= b (Substitute value of a)

⇒ c

^{zx}

^{}= b

⇒ (b

^{y})

^{zx}= b (Substituting value of c)

⇒ b

^{xyz}= b

^{1}

⇒ xyz = 1

**Q9: If 2**

^{x-1}+ 2^{x+1}= 320, find the value of x.Answer: Given, 2

^{x-1}+ 2

^{x+1}= 320

⇒ 2

^{-1}.2

^{x}+ 2

^{1}.2

^{x}= 320

⇒ 2

^{x}( 2

^{-1 }+ 2

^{1}) = 320

⇒ 2

^{x}(5/2) = 320

⇒ 2

^{x}= 320 x 2/5 = 128 = 2

^{7}

⇒ x = 7

**Q10: If a**

^{x-1}= bc, b^{y-1}= ca and c^{z-1}= ab then prove that xy + yz + zx = xyz.Answer: Given, a

^{x-1}= bc, b

^{y-1}= ca and c

^{z-1}= ab

a

^{x-1}= bc

⇒ a

^{x}/a = bc

⇒ a

^{x}= abc

Similarly, b

^{y}= abc and c

^{xz}= abc

Or we can write these terms as,

a

^{x}= abc ⇒ a = (abc)

^{1/x}

b

^{y}= abc ⇒ b = (abc)

^{1/y}

c

^{z}= abc ⇒ c = (abc)

^{1/z}

Multiplying above three equations give,

abc = (abc)

^{1/x}. (abc)

^{1/y}. (abc)

^{1/z}

⇒ abc = (abc)

^{1/x + 1/y + 1/z}

⇒ 1 = 1/x + 1/y + 1/z

⇒ 1 = (yz + xz + xy)/xyz

⇒ xyz = xy + yz + zx

**Q11: If x, y are real numbers such that 3**

^{(x/y +1)}- 3^{(x/y -1)}= 24, the find the value of (x+y)/(x-y).Answer: 3

^{(x/y +1)}- 3

^{(x/y -1)}= 24

⇒ 3

^{1}.3

^{x/y}- 3

^{-1}.3

^{x/y}= 24

⇒ 3

^{x/y}(3

^{1}- 3

^{-1}) = 24

⇒ 3

^{x/y}(9-1)/3 = 24

⇒ 3

^{x/y}(8)/3 = 24

⇒ 3

^{x/y}= (24).(3/8)

⇒ 3

^{x/y}= ⇒ 3

^{2}

⇒ x/y = 2 = 2/1

Applying componendo and dividendo,

(x+y)/(x-y) = (2+1)/(2-1)

Thus (x+y)/(x-y) = 2

**Q12: What are like surds and unlike surds?**

Answer: Two surds

^{m}√a and

^{n}√b

**are like surds if m = n.**

In case m ≠ n, the surds are unequal.

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