+349 I have lost track of these things...
Anyway, here is a problem that isn't that hard, really.
GIVEN:
a and b are real numbers
\(a+b=10\)
\(a^2+b^2=44\)
Find \(a^3+b^3\)
Oh, and last time, the question was declared answered after a moderator aswered it, so if possible, please don't declare this question answered so than people can solve it.
I encourage everyone who reads this to not look at the other people's answer, so that you can solve it yourself.
+26393 GIVEN:
a and b are real numbers
\(a+b = 10 \\ a^2+b^2 = 44\)
Find \(a^3 + b^3\)
\(\begin{array}{|rcll|} \hline (a+b)^2 &=& a^2 + 2ab + b^2 \\ (a+b)^2 &=& (a^2 + b^2) +2ab \\ (10)^2 &=& (44) + 2ab \\ 100 &=& 44 + 2ab \quad & | \quad : 2 \\ 50 &=& 22 + ab \\ ab &=& 50-22 \\ \mathbf{ab} & \mathbf{=} & \mathbf{28} \\\\ (a^2+b^2)(a+b) &=& a^3 +a^2b + b^2a + b^3 \\ (a^2+b^2)(a+b) &=& a^3 +b^3 + ab(a+b) \\ 44\cdot 10 &=& a^3 +b^3 + 28\cdot 10 \\ 440 &=& a^3 +b^3 + 280 \\ a^3 +b^3 &=& 440 - 280 \\ \mathbf{ a^3 +b^3} & \mathbf{=} & \mathbf{160} \\ \hline \end{array}\)
a = 5 - i sqrt(3) ≈ 5.00000 - 1.73205 i and b = 5 + i sqrt(3) ≈ 5.00000 + 1.73205 i
a = 5 + i sqrt(3) ≈ 5.00000 + 1.73205 i and b = 5 - i sqrt(3) ≈ 5.00000 - 1.73205 i
Simplify the following:
(-(i sqrt(3)) + 5)^3 + (i sqrt(3) + 5)^3
(-(i sqrt(3)) + 5)^3 = (-(i sqrt(3)) + 5) (-(i sqrt(3)) + 5)^2:
(-(i sqrt(3)) + 5) (-(i sqrt(3)) + 5)^2 + (i sqrt(3) + 5)^3
(-(i sqrt(3)) + 5)^2 = 25 - 5 i sqrt(3) - 5 i sqrt(3) - 3 = 22 - 10 i sqrt(3):
(-(i sqrt(3)) + 5) -10 i sqrt(3) + 22 + (i sqrt(3) + 5)^3
(-i sqrt(3) + 5) (-10 i sqrt(3) + 22) = 5×22 + 5 (-10 i sqrt(3)) + -i sqrt(3)×22 + -i sqrt(3) (-10 i sqrt(3)) = 110 + -50 i sqrt(3) + -22 i sqrt(3) - 30 = -72 i sqrt(3) + 80:
-72 i sqrt(3) + 80 + (i sqrt(3) + 5)^3
(i sqrt(3) + 5)^3 = (i sqrt(3) + 5) (i sqrt(3) + 5)^2:
80 - 72 i sqrt(3) + (i sqrt(3) + 5) (i sqrt(3) + 5)^2
(i sqrt(3) + 5)^2 = 25 + 5 i sqrt(3) + 5 i sqrt(3) - 3 = 22 + 10 i sqrt(3):
80 - 72 i sqrt(3) + (i sqrt(3) + 5) 10 i sqrt(3) + 22
(i sqrt(3) + 5) (10 i sqrt(3) + 22) = 5×22 + 5×10 i sqrt(3) + i sqrt(3)×22 + i sqrt(3)×10 i sqrt(3) = 110 + 50 i sqrt(3) + 22 i sqrt(3) - 30 = 72 i sqrt(3) + 80:
80 - 72 i sqrt(3) + 72 i sqrt(3) + 80
80 - 72 i sqrt(3) + 80 + 72 i sqrt(3) = 160:
Answer: | 160
+349 #mindblown
#overcomplicated
#i'mimpressed
I don't understand that behemoth of a solution because I was too lazy to read it, but you got the right answer, except there is a much, MUCH simpler solution.
+26393 GIVEN:
a and b are real numbers
\(a+b = 10 \\ a^2+b^2 = 44\)
Find \(a^3 + b^3\)
\(\begin{array}{|rcll|} \hline (a+b)^2 &=& a^2 + 2ab + b^2 \\ (a+b)^2 &=& (a^2 + b^2) +2ab \\ (10)^2 &=& (44) + 2ab \\ 100 &=& 44 + 2ab \quad & | \quad : 2 \\ 50 &=& 22 + ab \\ ab &=& 50-22 \\ \mathbf{ab} & \mathbf{=} & \mathbf{28} \\\\ (a^2+b^2)(a+b) &=& a^3 +a^2b + b^2a + b^3 \\ (a^2+b^2)(a+b) &=& a^3 +b^3 + ab(a+b) \\ 44\cdot 10 &=& a^3 +b^3 + 28\cdot 10 \\ 440 &=& a^3 +b^3 + 280 \\ a^3 +b^3 &=& 440 - 280 \\ \mathbf{ a^3 +b^3} & \mathbf{=} & \mathbf{160} \\ \hline \end{array}\)
+476 a = 5 - i sqrt(3) ≈ 5.00000 - 1.73205 i and b = 5 + i sqrt(3) ≈ 5.00000 + 1.73205 i
a = 5 + i sqrt(3) ≈ 5.00000 + 1.73205 i and b = 5 - i sqrt(3) ≈ 5.00000 - 1.73205 i
Simplify the following:
(-(i sqrt(3)) + 5)^3 + (i sqrt(3) + 5)^3
(-(i sqrt(3)) + 5)^3 = (-(i sqrt(3)) + 5) (-(i sqrt(3)) + 5)^2:
(-(i sqrt(3)) + 5) (-(i sqrt(3)) + 5)^2 + (i sqrt(3) + 5)^3
(-(i sqrt(3)) + 5)^2 = 25 - 5 i sqrt(3) - 5 i sqrt(3) - 3 = 22 - 10 i sqrt(3):
(-(i sqrt(3)) + 5) -10 i sqrt(3) + 22 + (i sqrt(3) + 5)^3
(-i sqrt(3) + 5) (-10 i sqrt(3) + 22) = 5×22 + 5 (-10 i sqrt(3)) + -i sqrt(3)×22 + -i sqrt(3) (-10 i sqrt(3)) = 110 + -50 i sqrt(3) + -22 i sqrt(3) - 30 = -72 i sqrt(3) + 80:
-72 i sqrt(3) + 80 + (i sqrt(3) + 5)^3
(i sqrt(3) + 5)^3 = (i sqrt(3) + 5) (i sqrt(3) + 5)^2:
80 - 72 i sqrt(3) + (i sqrt(3) + 5) (i sqrt(3) + 5)^2
(i sqrt(3) + 5)^2 = 25 + 5 i sqrt(3) + 5 i sqrt(3) - 3 = 22 + 10 i sqrt(3):
80 - 72 i sqrt(3) + (i sqrt(3) + 5) 10 i sqrt(3) + 22
(i sqrt(3) + 5) (10 i sqrt(3) + 22) = 5×22 + 5×10 i sqrt(3) + i sqrt(3)×22 + i sqrt(3)×10 i sqrt(3) = 110 + 50 i sqrt(3) + 22 i sqrt(3) - 30 = 72 i sqrt(3) + 80:
80 - 72 i sqrt(3) + 72 i sqrt(3) + 80
80 - 72 i sqrt(3) + 80 + 72 i sqrt(3) = 160: SO THE ANSWER IS 160
