Surds and Indices

Answer: (a) 1

z = x^c = (y^a)^c = y^ac = (z^b)^ac = z^abc

∴ z¹ = z^abc ⇒ abc = 1

Answer: (b) √3

We have ( √(1+x) + √(1+x) )² = 1+x+1-x+2√(1+x)(1-x)

= 2 + 2√(1-x²)

= 2 + 2√(1 - 3/4)

= 2 + 2√(1/4)

= 2 + 2 . ½ = 2 + 1 = 3

∴ √(1+x) + √ (1-x) = √3

Answer: (b) √5 - √3

∴ Greatest is √5 - √3

Answer: (a) 1

2³ X 2² X 2^-5 = 2^3+2-5 = 2^5-5 = 2⁰ = 1

Answer: (c) ³√4

√2 = 2^½

⁶√3 = 3^1/6

³√4 = 4^1/3

⁴√5 = 5^¼

LCM of 2, 6, 3, 4 = 12

Now √2 = 2^½ = 2^6.(1/12) = (64)^1/12

⁶√3 = 3^1/6 = 3^2.(1/12) = 9^1/12

³√4 = 4^1/3 = 4^4.(1/12) = 256^1/12

⁴√5 = 5^¼ = 5^3.(1/12) = 125^1/12

Hence the greatest one is 256^1/12 i.e ³√4