Prove that x 3 + y 3 + 3 m = m 3 x 3 y 3 if x y x + y = m .

We have to prove that x 3 + y 3 + 3 m = m 3 x 3 y 3 if x y x + y = m . We have to cube both the sides of equation x y x + y = m using identity a + b 3 = a 3 + b 3 + 3 a b a + b . ⇒ x y x + y 3 = m 3 ⇒ x 3 y 3 x + y 3 = m 3 ⇒ x 3 y 3 ( x 3 + y 3 + 3 x y ( x + y ) ) = m 3 Place the value of x y x + y as m . ⇒ x 3 y 3 ( x 3 + y 3 + 3 m ) = m 3 Now transpose x 3 y 3 in denominator of m 3 . ⇒ ( x 3 + y 3 + 3 m ) = m 3 x 3 y 3 Hence, proved.

Prove that 27 x 3 + 1 8 x 3 = 189 if 2 x + 1 3 x = 4 .

We have to prove that 27 x 3 + 1 8 x 3 = 189 if 2 x + 1 3 x = 4 . We have to cube both the sides of equation 2 x + 1 3 x = 4 using identity a + b 3 = a 3 + b 3 + 3 a b a + b . ⇒ 2 x + 1 3 x 3 = 4 3 ⇒ 2 x 3 + 1 3 x 3 + 3 × 2 x × 1 3 x 2 x + 1 3 x = 64 ⇒ 8 x 3 + 1 27 x 3 + 2 2 x + 1 3 x = 64 Place the value of 2 x + 1 3 x as 4 . ⇒ 8 x 3 + 1 27 x 3 + 2 × 4 = 64 ⇒ 8 x 3 + 1 27 x 3 = 64 - 8 = 56 In order to shuffle the coefficient of x 3 as 27 and 8 we need to multiply both side with 27 8 . ⇒ 27 8 × 8 x 3 + 27 8 × 1 27 x 3 = 27 8 × 56 ⇒ 27 x 3 + 1 8 x 3 = 189 Hence, proved.

Find the value of a 3 - 1 a 3 + 2 if 2 a - 2 a + 1 = 0 .

We have to find the value of a 3 - 1 a 3 + 2 if 2 a - 2 a + 1 = 0 . We can write 2 a - 2 a + 1 = 0 as ⇒ 2 a - 2 a = - 1 Separate out the common element. ⇒ a - 1 a = - 1 2 We have to cube both the sides of equation a - 1 a = - 1 2 using identity x - y 3 = x 3 - y 3 - 3 x y x - y . ⇒ a - 1 a 3 = - 1 2 3 ⇒ a 3 - 1 a 3 - 3 × a × 1 a a - 1 a = - 1 8 ⇒ a 3 - 1 a 3 - 3 a - 1 a = - 1 Place the value of a - 1 a as - 1 2 . ⇒ a 3 - 1 a 3 - 3 × - 1 2 = - 1 8 ⇒ a 3 - 1 a 3 = - 1 8 - 3 2 = - 1 - 12 8 = - 13 8 = - 1 5 8 Hence, the required answer is - 1 5 8 .

E M B I B E

Ganitprabha (MATHEMATICS TEXT BOOK) Class 8>Chapter 5 - Cubes>Let's work out - 5.2

West Bengal Board Solutions for Chapter: Cubes, Exercise 5: Let's work out - 5.2

Author:West Bengal Board

West Bengal Board Mathematics Solutions for Exercise - West Bengal Board Solutions for Chapter: Cubes, Exercise 5: Let's work out - 5.2

Attempt the practice questions on Chapter 5: Cubes, Exercise 5: Let's work out - 5.2 with hints and solutions to strengthen your understanding. Ganitprabha (MATHEMATICS TEXT BOOK) Class 8 solutions are prepared by Experienced Embibe Experts.