Test: Surds And Indices- 1 - UPSC

Let (17)^3.5 x (17)^x = 17⁸
Then, (17)^{3.5 + x} = 17⁸
∴ 3.5 + x = 8
⇒ x = (8 - 3.5)
⇒ x = 4.5

Given: 
⇒ (a / b)^{x - 1} = (b / a)^{x - 3}
⇒ (a / b) ^{x - 1} = (b / a) ^{-(x - 3)} = (a / b)^{(3 - x)}

⇒ x - 1 = 3 - x
⇒ 2x = 4
⇒ x = 2

⇒  x^z = y² ⇔ 10^{(0.48 z)} = 10^{(2 x 0.70)} = 10^1.40
⇒ 0.48 z = 1.40
⇒ z = 140 / 48 = 35 / 12 = 2.9(approx.)

⇒ 5^a = 3125 ⇔ 5^a = 5⁵
⇒ a = 5.
∴ 5^{(a - 3)} = 5^{(5 - 3)} = 5² = 25.

⇒  3^x - y = 27 = 3³    ⇔    x - y = 3 ....(i)

⇒ 3^x + y = 243 = 3⁵    ⇔    x + y = 5 ....(ii)

On solving (i) and (ii), we get x = 4.

⇒  (256)^0.16 x (256)^0.09 = (256)^{(0.16 + 0.09)}

= (256)^0.25
= (256)^{(25 / 100)}
= (256)^{(1 / 4)}
= (4⁴)^{(1 / 4)}
= 4^{4(1 / 4)}
= 4¹
= 4

⇒ (10)¹⁵⁰ ÷ (10)¹⁴⁶ = 10¹⁵⁰ / 10¹⁴⁶
= (10)^150 -146
= 10⁴
= 10000.

Given Exp: 
1/(1 + x^b / x^a + x^c / x^a) + 1/(1 + x^a / x^b + x^c / x^b) + 1/(1 + x^b / x^c +x^a / x^c)

= x^a/(x^a + x^b + x^c ) + x^b/(x^a + x^b + x^c ) + x^c/(x^a + x^b + x^c )
= (x^a + x^b + x^c ) / (x^a + x^b + x^c )
= 1.

Let (25)^7.5 x (5)^2.5 ÷ (125)^1.5 = 5^x.
Then, (5²)^7.5 x 5^2.5 / (5³) ^1.5 = 5^x

⇒ (5²)^7.5 x 5^2.5 / (5)^3 x 1.5 = 5^x
⇒ 5¹⁵ x 5^2.5 / 5^4.5 = 5^x
⇒ 5^x = 5^{(15 + 2.5 - 4.5)}
⇒ 5^x = 5¹³
∴ x = 13.

⇒ (0.04)^-1.5 = (4 / 100)^-1.5

= (1 / 25) ^{-(3 / 2)}
= (5²)^{(3 / 2)}
= (5)^{2 x (3 / 2)}
= 5³
= 125.