Question

If a + b = 5 and ab = 6, then find the value of $$a^{3}+b^{3}$$.

Answer 1

Hint:In this question the given values are a + b = 5 and ab = 6, and we have to find the value of $$a^{3}+b^{3}$$. So to find the solution we have to use one identity, that is,
$$a^{3}+b^{3}=\left( a+b\right)^{3} -3ab\left( a+b\right) $$......(1)
So by using the above identity we can able to find the solution.

Complete step by step answer:
Given,
a + b = 5……………...(2)
ab = 6 ………………..(3)
Now by using the identity (1), we can write,
$$a^{3}+b^{3}$$
$$=\left( a+b\right)^{3} -3ab\left( a+b\right) $$
Now by putting the value of (a + b) and ab in the above equation, we get,
$$a^{3}+b^{3}=\left( 5\right)^{3} -3\times 6\times 5$$
$$\Rightarrow a^{3}+b^{3}=(5\times 5\times 5)-(3\times 6\times 5)$$
$$\Rightarrow a^{3}+b^{3}=125-90$$
$$\Rightarrow a^{3}+b^{3}=35$$
Therefore the value of $$a^{3}+b^{3}$$ is 35.

Note:
 So you might be thinking why we use this formula (1), because in the question the given values are (a + b) and ab so if we apply this formula then we can easily put their value, which helps us to solve it in an easier way.
Also you forget the formula(1) while solving then you can establish the formula,
So as we know that,
$$\left( a+b\right)^{3} =a^{3}+3a^{2}b+3ab^{2}+b^{3}$$
From RHS you can write the value of $$a^{3}+b^{3}$$
$$a^{3}+3a^{2}b+3ab^{2}+b^{3}=\left( a+b\right)^{3} $$ [by reversing the equation]
$$\Rightarrow a^{3}+b^{3}=\left( a+b\right)^{3} -3a^{2}b-3ab^{2}$$ [taking the remaining term left side]
$$\Rightarrow a^{3}+b^{3}=\left( a+b\right)^{3} -3ab\left( a+b\right) $$
By taking common (-3ab) from the second and third term.