A mixture contains alcohol and water in the ratio 4 : 3. If 5 liters of water is added to the mixture, the ratio becomes 4: 5. Find the quantity of alcohol in the given mixture.

176 = 2 x 2 x 2 x 2 x 11
So, in order to make it a perfect square, it must be multiplied by 11.

Number of members = ?2209
= 47

?256 / ?x = 2
? 16 = 2?x
? ?x = 8
? x = 64.

N = 25/16
? ?N = ?25/16
= 5/4
= 1¹/₄

(x + 1)/(x - 1) - (x - 1)/(x + 1) = {(x + 1)² - (x - 1)²} / {(x - 1) (x + 1)}
= (x² + 1 + 2x) - (x² + 1 - 2x) / (x² - 1)
= 4x / (x² - 1) = 4(?3 + ?2) / {(?3 + ?2)² - 1}
= 4(?3 + ?2) / (3 + 2 + 2?6 - 1) = 4(?3 + ?2) / (4 + 2 ?6)
= 4(?3 + ?2) / 2 ?2 (?3 + ?2)
= ?2

?8 and ?1000 = 8^1/3 and 1000^1/3
Since 1000 > 8
? ?1000 > ?8

?81 and ?1 = 81^1/2 and 1^1/2
Since, 1 < 81
? ?1 < ?81

Let the number be x.
According to the question,
x² + (14)³ = 4425
? x² = 4425 - (14)³
? x² = 4425 - 2744
? x² = 1681
? x = ?1681 = 41

?10 and ?8 = 10^1/4 and 8^1/3
Since, LCM of 4 and 3 is 12.
? 10^1/4 = 10^3/12 = (10³)^1/12 = (1000)^1/12 = ¹²?1000
and 8^1/3 = 8^4/12 = (8⁴)^1/12 (4096)^1/2 = ¹²?4096
? 4096 > 1000
Hence, 8^1/3 >10^1/4
? ?8 > ?10

74088 = 2 x 2 x 2 x 3 x 3 x 3 x 7 x 7 x 7
= (2 x 2) x (3 x 3) x (7 x 7) x ( 2 x 3 x 7)
? Least number which should be multiplied to 74088 to make it a perfect suqare = 2 x 3 x 7 = 42