Question

14 If 2a - 3b = 3 and ab = 2, find the value of 8a3-27b3​

Answer 1

Step-by-step explanation:

Given:-

2a - 3b = 3

and ab = 2

To find:-

Find the value of 8a^3-27b^3 ?

Solution:-

Given that

2a - 3b = 3 --------(1)

and ab = 2--------(2)

On cubing both sides of the equation (1)

=>(2a-3b)^3 = (3)^3

LHS is in the form of (a-b)^3

Where , a= 2a and b= 3b

We know that

(a-b)^3 = a^3-3a^2b+3ab^2-b^3

=>(a-b)^3=a^3-3ab(a-b)-b^3

On applying this formula for (2a-3b)^3

(2a)^3-3(2a)(3b)(2a-3b)-(3b)^3

therefore (2a-3b)^3 = (3)^3

=> (2a)^3-3(2a)(3b)(2a-3b)-(3b)^3 = 3^3

=> 8a^3-18ab(2a-3b)-27b^3 = 27

=> 8a^3-18ab(3) -27b^3 = 27

(from (1))

=>8a^3-54ab-27b^3 = 27

=>8a^3-54(2)-27b^3 = 27

(from (2))

=> 8a^3-27b^3 -108= 27

=> 8a^3-27b^3 = 27+108

=> 8a^3 -27b^3 = 135

Answer:-

The value of 8a^3-27b^3 for the given problem is 135

Used formulae:-

• (a-b)^3 = a^3-3a^2b+3ab^2-b^3

• (a-b)^3=a^3-3ab(a-b)-b^3