FORMULA FOR A CUBE PLUS B CUBE

Problem 1:

If a + b = 3 and ab = 2, find the value of

(a) a² - ab + b² and (b) a³ + b³

Solution :

a + b = 3 and ab = 2

a³ + b³ = (a + b) (a² - ab + b²)

By applying the given values, we get

a³ + b³ = 3 (a² - 2 + b²)

Here, a² + b² = (a + b)² - 2ab

a² + b² = 3² - 2(2)

= 9 – 4

a² + b² = 5

Applying a² + b² = 5

(a)  a² - ab + b²

a² + b² - ab = 5 – 2

a² + b² - ab = 3

(b)  a³ + b³

a³ + b³ = 3(5 - 2)

= 3(3)

a³ + b³ = 9

Problem 2 :

If a - b = 5 and ab = 36, find the value of

(a) a² + ab + b² and (b) a³ - b³

Solution :

a - b = 5 and ab = 36

a³ - b³ = (a - b) (a² + ab + b²)

a³ - b³ = 5 (a² + 36 + b²)

Here, a² + b² = (a - b)² + 2ab

a² + b² = 5² + 2(36)

= 25 + 72

a² + b² = 97

Applying a² + b² = 97

(a) a² + ab + b²

a² + b² + ab = 97 + 36

a² + b² + ab = 133

(b)  a³ - b³

a³ - b³ = 5(97 + 36)

a³ - b³ = 5(133)

a³ - b³ = 665

Problem 3 :

If m + 1/m = a, find the value of m³ + 1/m³.

Solution :

Given m + 1/m = a

(m + 1/m)³ = a³

Use the formula (a + b)³ = a³ + b³ + 3ab(a + b)

m³ + 1/m³ + 3(m)(1/m) (m + 1/m) = a³

m³ + 1/m³ + 3m + 3/m = a³

m³ + 1/m³ = a³ - 3(m + 1/m)

= a³ - 3(a)

m³ + 1/m³ = a³ - 3a

Problem 4 :

If x - 1/x = p, find the value of x³ - 1/x³.

Solution :

Given x - 1/x = p

(x - 1/x)³ = p³

Use the formula (a - b)³ = a³ - b³ - 3ab(a - b)

x³ - 1/x³ - 3(x)(1/x) (x - 1/x) = p³

x³ - 1/x³ - 3x + 3/x = p³

x³ - 1/x³ = p³ + 3(x - 1/x)

= p³ + 3(p)

x³ - 1/x³ = p³ + 3p

Problem 5 :

If a - 1/a = 1, show that, a³ - 1/a³ = 4.

Solution :

Given a - 1/a = 1

 (a - 1/a)² = 1²

Use the formula (a - b)² = a² - 2ab + b²

a² - 2(a)(1/a) + (1/a)² = 1

a² - 2 + 1/a² = 1

a² + 1/a² = 3

So, a³ - 1/a³ = a³ - (1/a)³

Use the formula (a³ - b³) = (a - b) (a² + ab + b²)

= (a - 1/a) [a² + a∙1/a + (1/a)²]

= (a - 1/a) (a² + 1 + 1/a²)

= (1) (3 + 1)

= 1 × 4

= 4

So, a³ - 1/a³ = 4.

Problem 6 :

If 2x - 2/x = 3, show that, 8(x³ - 1/x³) = 63.

Solution :

Given 2x - 2/x = 3

2(x – 1/x) = 3

x - 1/x = 3/2

(x - 1/x)³ = (3/2)³

(x - 1/x)³ = 27/8

(x - 1/x)² (x - 1/x) = 27/8

Use the formula (a - b)² = a² - 2ab + b²

(x² - 2 + 1/x²) (x - 1/x) = 27/8

x²(x) - x²(1/x) – 2(x) + 2(1/x) + 1/x²(x) – 1/x²(1/x) = 27/8

x³ - x - 2x + 2/x + 1/x – 1/x³ = 27/8

(x³ - 1/x³) – (x – 1/x) – (2x – 2/x) = 27/8

x³ - 1/x³ = 27/8 + 3 + 3/2

x³ - 1/x³ = (27 + 24 + 12) / 8

x³ - 1/x³ = 63/8

So, 8(x³ - 1/x³) = 8(63/8) = 63.