if $a+ b = 9 $and $ab = 1$. what will be the $a^3 + b^3 =$?

Question

How can I solve this? Or, is it given properly? If $a + b = 9$ and $ab = 1$. What is $a^3 + b^3 = $?

Answer 1

$a + b = 9 => a = 9 - b$ and then you solve $b$: $ab = 1 => (9 - b)b = 1$; solving $a$ is easy now. (then $a^3 + b^3 = ...$ is easier)

Answer 2

You can start with the fact that $$a^3+b^3=(a+b)(a^2-ab+b^2) $$ $$(a+b)^2 = a^2+2ab+b^2$$ And see if you can use some substitutions, additions and subtractions.

Answer 3

Hint: $a^3 + b^3 = (a+b)((a+b)^2 - 3ab)$

Answer 4

Hint: Express $a^3+b^3$ as function of $ab$ and $a+b$.

Answer 5

$$a^2+b^2=(a+b)^2-2ab$$ $$a^3+b^3=(a+b)^3-3ab(a+b)$$ $$a^4+b^4=(a+b)^4-4ab(a+b)^2+2a^2b^2$$ $$a^5+b^5=(a+b)^5-5ab(a+b)^3+5a^2b^2(a+b)$$