if a+b=3 and a^2+b^2=6, what is a^3 + b^3
To find the value of a^3 + b^3, we can use the following identity:
a^3 + b^3 = (a + b) * (a^2 - ab + b^2)
From the given information, we know that a + b = 3 and a^2 + b^2 = 6. We can substitute these values into the above identity to find the answer.
Substituting a + b = 3 into the identity:
a^3 + b^3 = (3) * (a^2 - ab + b^2)
Now, let's simplify using the second equation given:
a^2 + b^2 = 6
Multiplying the first equation by (a - b):
(a + b) * (a^2 - ab + b^2) = 3 * (6 - ab)
Expanding the left side:
a^3 + b^3 + ab(a - b) = 18 - 3ab
From this equation, we need to find the value of ab to proceed. Let's use the second given equation to find it.
a^2 + b^2 = 6
Squaring the first equation:
(a + b)^2 = 9
Expanding the left side:
a^2 + 2ab + b^2 = 9
Substituting a^2 + b^2 = 6 into the equation:
6 + 2ab = 9
Rearranging the equation:
2ab = 3
Dividing both sides by 2 to solve for ab:
ab = 3/2
Now, substituting the value of ab back into the equation we left earlier:
a^3 + b^3 + ab(a - b) = 18 - 3ab
a^3 + b^3 + ab(a - b) = 18 - 3(3/2)
a^3 + b^3 + ab(a - b) = 18 - 9/2
a^3 + b^3 + ab(a - b) = 27/2
So, the value of a^3 + b^3 is 27/2.