[tex]\large\underline{\sf{Solution-}}[/tex]
Given that,
[tex]\rm \: a + b = 7 \\ [/tex]
and
[tex]\rm \: ab = 12 \\ [/tex]
Now, Consider
On cubing both sides, we get
[tex]\rm \: {(a + b)}^{3} = {(7)}^{3} \\ [/tex]
[tex]\rm \: {a}^{3} + {b}^{3} + 3ab(a + b) = 343 \\ [/tex]
On substituting the values of a + b and ab, we get
[tex]\rm \: {a}^{3} + {b}^{3} + 3 \times 12 \times 7 = 343 \\ [/tex]
[tex]\rm \: {a}^{3} + {b}^{3} + 252 = 343 \\ [/tex]
[tex]\rm \: {a}^{3} + {b}^{3} = 343 - 252 \\ [/tex]
[tex]\bf\implies \: {a}^{3} + {b}^{3} = 91 \\ \\ [/tex]
[tex]\rule{190pt}{2pt} \\ [/tex]
Additional Information :-
[tex]\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}[/tex]
• a+b = 7
• ab = 12
The value of [tex] {a}^{3} + {b}^{3} [/tex]
Given that
• a+b = 7 -------(1)
• ab = 12 -------(2)
On squaring both sides of the equation (1) then
(a+b)² = 7²
=> a²+2ab + b² = 49
=> a²+b² +2 (12) = 49
=> a²+b² + 24 = 49
=> a²+b² = 49-24
=> a²+b² = 25 --------(3)
We know that
a³+b³ = (a+b)(a²-ab+b²)
=> a³+b³ = (a+b)(a²+b²-ab)
=> a³+b³ = (7)(25-12)
=> a³+b³ = (7)(13)
=> a³+b³ = 91
a³+b³ = (a+b)[(a+b)²-3ab]
=> a³+b³ = (7)[(7)²-3(12)]
=> a³+b³ = (7)(49-36)
a³+b³ = (a+b)³-3ab(a+b)
=> a³+b³ = (7)³-3(12)(7)
=> a³+b³ = 343-252
• The value of [tex] {a}^{3} + {b}^{3} [/tex] is 91
• (a+b)² = a²+2ab+b²
• a³+b³ = (a+b)(a²-ab+b²)
• a³+b³ = (a+b)[(a+b)²-3ab]
• a³+b³ = (a+b)³ - 3ab(a+b)
Copyright © 2024 EHUB.TIPS team's - All rights reserved.
Answers & Comments
Verified answer
[tex]\large\underline{\sf{Solution-}}[/tex]
Given that,
[tex]\rm \: a + b = 7 \\ [/tex]
and
[tex]\rm \: ab = 12 \\ [/tex]
Now, Consider
[tex]\rm \: a + b = 7 \\ [/tex]
On cubing both sides, we get
[tex]\rm \: {(a + b)}^{3} = {(7)}^{3} \\ [/tex]
[tex]\rm \: {a}^{3} + {b}^{3} + 3ab(a + b) = 343 \\ [/tex]
On substituting the values of a + b and ab, we get
[tex]\rm \: {a}^{3} + {b}^{3} + 3 \times 12 \times 7 = 343 \\ [/tex]
[tex]\rm \: {a}^{3} + {b}^{3} + 252 = 343 \\ [/tex]
[tex]\rm \: {a}^{3} + {b}^{3} = 343 - 252 \\ [/tex]
[tex]\bf\implies \: {a}^{3} + {b}^{3} = 91 \\ \\ [/tex]
[tex]\rule{190pt}{2pt} \\ [/tex]
Additional Information :-
[tex]\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}[/tex]
Given :-
• a+b = 7
• ab = 12
To find :-
The value of [tex] {a}^{3} + {b}^{3} [/tex]
Solution :-
Given that
• a+b = 7 -------(1)
• ab = 12 -------(2)
On squaring both sides of the equation (1) then
(a+b)² = 7²
=> a²+2ab + b² = 49
=> a²+b² +2 (12) = 49
=> a²+b² + 24 = 49
=> a²+b² = 49-24
=> a²+b² = 25 --------(3)
We know that
a³+b³ = (a+b)(a²-ab+b²)
=> a³+b³ = (a+b)(a²+b²-ab)
=> a³+b³ = (7)(25-12)
=> a³+b³ = (7)(13)
=> a³+b³ = 91
Alternative Method:-
We know that
a³+b³ = (a+b)[(a+b)²-3ab]
=> a³+b³ = (7)[(7)²-3(12)]
=> a³+b³ = (7)(49-36)
=> a³+b³ = (7)(13)
=> a³+b³ = 91
or
a³+b³ = (a+b)³-3ab(a+b)
=> a³+b³ = (7)³-3(12)(7)
=> a³+b³ = 343-252
=> a³+b³ = 91
Answer :-
• The value of [tex] {a}^{3} + {b}^{3} [/tex] is 91
Used formulae:-
• (a+b)² = a²+2ab+b²
• a³+b³ = (a+b)(a²-ab+b²)
• a³+b³ = (a+b)[(a+b)²-3ab]
• a³+b³ = (a+b)³ - 3ab(a+b)