1 Marks Question

Question 11 Mark

Write the value of $30^3 + 20^3 - 50^3.$

Answer

The given expression is
$30^3 + 20^3 - 50^3$
Let $a = 30, b = 20$ and $c = -50.$ Then the given expression becomes $30^3 + 20^3 - 50^3$
$ = a^3 + b^3 + c^3$
Note that
$a + b + c = 30 + 20 + (-50)$
$= 30 + 20 - 50$
$= 0$
Recall the formula
$a^3 + b^3 + c^3 - 3abc$
$= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
when$ a + b + c = 0,$ this becomes
$a^3 + b^3 + c^3 - 3abc$
$= 0.(a^2 + b^2 + c^2 - ab - bc - ca)$
$= 0$
$a^3 + b^3 + c^3 = 3abc$
So, we have the new formula
$a^3 + b^3 + c^3 = 3abc,$ when $a + b + c = 0$
Using the above formula, the value of the given expression is
$a^3 + b^3 + c^3 = 3abc$
$30^3 + 20^3 - 50^3 $
$= 3.(30).(20).(-50)$
$30^3 + 20^3 - 50^3 $
$= -90000$

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Question 21 Mark

Write the value of $48^3 - 30^3 - 18^3.$

Answer

The given expression is
$48^3 - 30^3 - 18^3$
Let $a = 48, b = -30$ and $c = -18$. Then the given expression becomes $48^3 - 30^3 - 18^3$
$= a^3 + b^3 + c^3$
Note that
$a + b + c = 48 + (-30) + (-18)$
$= 48 - 30 - 18$
$= 0$
Recall the formula
$a^3 + b^3 + c^3 - 3abc$
$= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
when $a + b + c = 0,$ this becomes
$a^3 + b^3 + c^3 - 3abc$
$ = 0.(a^2 + b^2 + c^2 - ab - bc - ca)$
$= 0$
$a^3 + b^3 + c^3 = 3abc$
So, we have the new formula
$a^3 + b^3 + c^3 = 3abc$, when $a + b + c = 0$
Using the above formula, the value of the given expression is
$a^3 + b^3 + c^3 = 3abc$
$48^3 - 30^3 - 18^3 = 3.(48).(-30).(-18)$
$48^3 - 30^3 - 18^3 = 77760$

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Question 31 Mark

Write the value of $\Big(\frac{1}{2}\Big)^3+\Big(\frac{1}{3}\Big)^3-\Big(\frac{5}{6}\Big)^3.$

Answer

The given expression is
$\Big(\frac{1}{2}\Big)^3+\Big(\frac{1}{3}\Big)^3-\Big(\frac{5}{6}\Big)^3$
Let $\text{a}=\frac{1}{2},\text{b}=\frac{1}{3}$ and $\text{c}=-\frac{5}{6}.$
Then the given expression becomes $\Big(\frac{1}{2}\Big)^3+\Big(\frac{1}{3}\Big)^3-\Big(\frac{5}{6}\Big)^3$
$=\text{a}^3+\text{b}^3+\text{c}^3$
Note that
$\text{a}+\text{b}+\text{c}=\frac{1}{2}+\frac{1}{3}+\Big(\frac{5}{6}\Big)$
$=\frac{1}{2}+\frac{1}{3}-\frac{5}{6}$
$=0$
Recall the formula
$a^3 + b^3 + c^3 - 3abc $
$= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
when $a + b + c = 0,$ this becomes
$a^3 + b^3 + c^3 - 3abc$
$ = 0.(a^2 + b^2 + c^2 - ab - bc - ca)$
$= 0$
$a^3 + b^3 + c^3 = 3abc$
So, we have the new formula
$a^3 + b^3 + c^3 = 3abc,$ when$ a + b + c = 0$
Using the above formula, the value of the given expression is
$a^3 + b^3 + c^3 = 3abc$
$\Big(\frac{1}{2}\Big)^3+\Big(\frac{1}{3}\Big)^3-\Big(\frac{5}{6}\Big)^3$
$=3.\Big(\frac{1}{2}\Big).\Big(\frac{1}{3}\Big).\Big(-\frac{5}{6}\Big)$
$\Big(\frac{1}{2}\Big)^3+\Big(\frac{1}{3}\Big)^3-\Big(\frac{5}{6}\Big)^3$
$=\Big(-\frac{5}{12}\Big)$

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Question 41 Mark

Write the value of $25^3 - 75^{3 }+ 50^3.$

Answer

The given expression is
$25^3 - 75^{3 }+ 50^3$
Let $a = 25, b = -75$ and $c = 50$. Then the given expression becomes $25^3 - 75^{3 }+ 50^3$
$ = a^3 + b^3 + c^3$
Note that
$a + b + c = 25 + (-75) + 50$
$= 25 - 75 + 50$
$= 0$
Recall the formula
$a^3 + b^3 + c^3 - 3abc$
$= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
when $a + b + c = 0$, this becomes
$a^3 + b^3 + c^3 - 3abc$
$ = 0.(a^2 + b^2 + c^2 - ab - bc - ca)$
$= 0$
$a^3 + b^3 + c^3 = 3abc$
So, we have the new formula
$a^3 + b^3 + c^3 = 3abc,$ when $a + b + c = 0$
Using the above formula, the value of the given expression is
$a^3 + b^3 + c^3 = 3abc$
$25^3 - 75^{3 }+ 50^3 = 3.(25).(-75).(50)$
$25^3 - 75^{3 }+ 50^3 = - 281250$

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