Question

If a+b=8 and ab=12, find the value of a³+b³​

Answer 1

Answer:

224

solution:

a + b = 8

ab = 12

i) a + b = 8

2 + 6 = 8

ii) ab = 12

a × b = 12

2 × 6 = 12

iii)a^3 + 6^3

2^3 + 6^3

8 + 216 = 224

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: ab = 12$

and

$\rm \: a + b = 8$

On cubing both sides, we get

$\rm \: {(a + b)}^{3} = {8}^{3}$

$\rm \: {a}^{3} + {b}^{3} + 3ab(a + b) = 512$

can be further rewritten as on substituting the values of ab and a + b,

$\rm \: {a}^{3} + {b}^{3} + 3 \times 12 \times 8 = 512$

$\rm \: {a}^{3} + {b}^{3} + 288 = 512$

$\rm \: {a}^{3} + {b}^{3} = 512 - 288$

$\rm\implies \:\boxed{ \rm{ \: {a}^{3} + {b}^{3} = 224 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$