Question

if a+b+c=9 and ab+bc+ca=23 find a^3+b^3+c^3

Answer 1

ANSWER:

• Value of a³+b³+c³ = 108.

GIVEN:

• a+b+c = 9
• ab+bc+ca = 23 .....(i)

TO FIND:

• Value of a³+b³+c³

SOLUTION;

=> a+b+c = 9

Squaring both sides we get:

=> (a+b+c)² = (9)²

=> a²+b²+c²+2(ab+bc+ca) = 81

=> a²+b²+c²+2(23) = 81. [From ..(i)]

=> a²+b²+c² = 81-46

=> a²+b²+c² = 35

Now formula:

• a³+b³+c³ = (a+b+c)[a²+b²+c²-(ab+bc+ca)]

Putting the values in the formula:

=> a³+b³+c³ = 9(35-23)

=> a³+b³+c³ = 9(12)

=> a³+b³+c³ = 108.

Value of a³+b³+c³ = 108.

NOTE:

Some important formulas:

(a+b)² = a²+b²+2ab

(a-b)² = a²+b²-2ab

(a+b)(a-b) = a²-b²

(a+b)³ = a³+b³+3ab(a+b)

(a-b)³ = a³-b³-3ab(a-b)

a³+b³ = (a+b)(a²+b²-ab)

a³-b³ = (a-b)(a²+b²+ab)

(a+b)² = (a-b)²+4ab

(a-b)² = (a+b)²-4ab