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Find the value of $a^3+8b^3$.

Last updated date: 19th Jul 2024
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Hint: We are given two equations in $a$ and $b$. And we are required to find the value of an expression which is also in terms of $a$ and $b$. For this, we can use two methods. We can individually find the value of both $a$ and $b$ by substituting the value of one in terms of another and putting it in the second equation. Or we can use the cubic identity and do this using the short trick.

Complete step by step answer:
We have$a+2b=10$
We need the involvement of the product $ab$ as well. So, we are going to use the following identity here:
For any two numbers $x$ and $y$, the following holds true:
So, we take cube of both sides of the equation below:
$\implies \left(a+2b\right)^3=10^3=1000$
Using the identity, we open the left hand side to get:
$a^3+8b^3+3\times a\times 2b\left(a+2b\right)=1000$
$a^3+8b^3+6ab\times 10=1000$
This happens because the value of $a+2b$ is given to be 10.
Also, $ab=15$. We put this in the equation:
$a^3+8b^3+6\times 15\times 10=1000$
Hence, the value of the expression has been found out to be 100.

Note: Since we are given two equations in two variables, we can plug the value of one into another equation by expressing one in terms of another i.e. we can put $a=10-2b$ and then since $ab=15$, we would get:
$\left(10-2b\right)\times b=15$
$\implies 2b^2-10b+15=0$
And then we can solve this quadratic equation to obtain the value of $b$ and then back substitute it to find the value of $a$. But this becomes a really long process and also increases the chance of calculation mistakes. So, try to create identities in such questions so that you get the most accurate answer in the least amount of time.