If 5 a . 125 a + 625 a 25 - b . 8 1 3 = 125 , show that 4 a + 2 b = 3 .

Given, 5 a . 125 a + 625 a 25 - b . 8 1 3 = 125 To prove that: 4 a + 2 b = 3 We need to simplify the given expression. We have, 5 a . 125 a + 625 a 25 - b . 8 1 3 = 125 ⇒ 5 a × 5 3 a + 5 4 a 5 2 - b × 2 3 1 3 = 5 3 ⇒ 5 a × 5 3 a + 5 4 a 5 - 2 b × 2 3 × 1 3 = 5 3 ∴ a n m = a n m ⇒ 5 a + 3 a + 5 4 a 5 - 2 b × 2 = 5 3 ∴ a n × a m = a n + m ⇒ 5 4 a + 5 4 a 5 - 2 b × 2 = 5 3 ⇒ 5 4 a 1 + 1 5 - 2 b × 2 = 5 3 ⇒ 5 4 a × 2 5 - 2 b × 2 = 5 3 ⇒ 5 4 a 5 - 2 b = 5 3 ⇒ 5 4 a - ( - 2 b ) = 5 3 ∴ a n ÷ a m = a n - m ⇒ 5 4 a + 2 b = 5 3 Equating the powers with same bases. 4 a + 2 b = 3 Hence, it is proved.

If a = 10 3 × 0 . 0048 and b = 10 - 1 × 75 , find the value of a b and a b .

Given, a = 10 3 × 0 . 0048 and b = 10 - 1 × 75 We have to find the value of a b and a b . By substituting the value of a and b in a b . Thus, a b = 10 3 × 0 . 0048 × 10 - 1 × 75 = 10 3 - 1 × 0 . 0048 × 75 = 10 2 × 0 . 0048 × 75 ∴ a n × a m = a n + m = 100 × 0 . 0048 × 75 a b = 0 . 48 × 75 = 36 = 6 So, the value of a b = 6 a b = 10 3 × 0 . 0048 10 - 1 × 75 = 10 3 - ( - 1 ) × 0 . 0048 75 ∴ a n ÷ a m = a n - m = 10 4 × 0 . 0048 75 = 10000 × 0 . 0048 75 = 48 75 = 48 75 ÷ 3 3 a b = 16 25 = 4 5 So, the value of a b = 4 5 . Therefore, the value of a b is 6 and a b is 4 5 .

Solve for x : 9 25 1 - 2 x = 125 27 x - 2

Given, 9 25 1 - 2 x = 125 27 x - 2 We have to find the value of x in given expression. Thus, 9 25 1 - 2 x = 125 27 x - 2 3 2 5 2 1 - 2 x = 5 3 3 3 x - 2 3 5 2 1 - 2 x = 5 3 3 x - 2 ∴ a n b n = a b n 3 5 2 - 4 x = 5 3 3 x - 6 ∴ a n m = a n m 5 3 - 2 - 4 x = 5 3 3 x - 6 ∴ a b n = b a - n 5 3 4 x - 2 = 5 3 3 x - 6 Equating the powers with same bases. 4 x - 2 = 3 x - 6 4 x - 3 x = - 6 + 2 x = - 4 Therefore, the value of x in 9 25 1 - 2 x = 125 27 x - 2 is - 4 .