Without actually calculating find the value of 25 cube minus 75 cube + 50 cube
Answers
\begin{gathered}To \: find \: = > The \: value \: of \: {25}^{3} - {75}^{3} + {50}^{3} \: without \: doing \: actual \: calculations. \\ \\ As \: we \: know \: that \: \\ \\ {a}^{3} + {b}^{3} + {c}^{3} = 3abc \\ if \: a + b + c = 0 \\ \\ Here \: let \: a = 25 \\ b = - 75 \\ c = 50 \\ \\ now \\ \\ 25 + ( - 75) + 50 \\ \\ = > 25 - 75 + 50 \\ \\ = > 75 - 75 \\ \\ = > \: 0\end{gathered}
Tofind=>Thevalueof25
3
−75
3
+50
3
withoutdoingactualcalculations.
Asweknowthat
a
3
+b
3
+c
3
=3abc
ifa+b+c=0
Hereleta=25
b=−75
c=50
now
25+(−75)+50
=>25−75+50
=>75−75
=>0
\begin{gathered}As \: if \: a + b + c = 0 \: then \: {a}^{3} + {b}^{3} + {c}^{3} = 3abc \\ \\ Therefore \: we \: can \: say \: that \: \\ \\ {25}^{3} + ( - {75)}^{3} + {50}^{3} = 3(25)( - 75)(50) \\ \\ = > 3 \times 25 \times ( - 75) \times 50 \\ \\ = > - (3 \times 25 \times 75 \times 50) \\ \\ = > \: - (150 \times 25 \times 75) \\ \\ = > - (3750 \times 75) \\ \\ = > - 281250 \\ \\ Therefore \: \: {25}^{3} - {75}^{3} + {50}^{3} = - 281250\end{gathered}
Asifa+b+c=0thena
3
+b
3
+c
3
=3abc
Thereforewecansaythat
25
3
+(−75)
3
+50
3
=3(25)(−75)(50)
=>3×25×(−75)×50
=>−(3×25×75×50)
=>−(150×25×75)
=>−(3750×75)
=>−281250
Therefore25
3
−75
3
+50
3
=−281250