Question

using suitable identity find (17)^3-(12)^3-(5)^3. pls it's very urgent.​

Answer 1

We may recall the identity given below

$\large(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=a^3+b^3+c^3-3abc$

Here,

$a = 17\\ \\ b=-12\\ \\ c=-5$

And we see that,

$a+b+c=17-12-5=0$

So,

$\begin{aligned}&a+b+c=0\\ \\ \Longrightarrow\ \ &(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=0\\ \\ \Longrightarrow\ \ &a^3+b^3+c^3-3abc=0\\ \\ \Longrightarrow\ \ &a^3+b^3+c^3=3abc\end{aligned}$

Now we get a simpler method. By using this,

$(17)^3-(12)^3-(5)^3\ =\ 3\cdot (17)\cdot (-12)\cdot (-5)\ =\ \mathbf{3060}$

Hence found by using suitable identity!

Let's check!

$(17)^3-(12)^3-(5)^3=4913-1728-125=\mathbf{3060}$

Hence checked and it's true!