Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \:(1) \: \: \dfrac{ {8}^{3m} \times {9}^{3n} \times {(125)}^{2p} }{ {(16)}^{2m} \times {(27)}^{n} \times {(25)}^{3p} } = {2}^{m} \times {3}^{3n}\qquad \: \\ \\& \qquad \:\sf \: (2) \: \: {7x}^{2} + 14xy + {7y}^{2} = 700\end{aligned}} \qquad \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given expression is
[tex]\sf \: \dfrac{ {8}^{3m} \times {9}^{3n} \times {(125)}^{2p} }{ {(16)}^{2m} \times {(27)}^{n} \times {(25)}^{3p} } \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \dfrac{ {( {2}^{3}) }^{3m} \times {( {3}^{2}) }^{3n} \times {( {5}^{3} )}^{2p} }{ {( {2}^{4} )}^{2m} \times {( {3}^{3} )}^{n} \times {( {5}^{2} )}^{3p} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{ {2}^{9m} \times {3}^{6n} \times {(5)}^{6p} }{ {(2)}^{8m} \times {(3)}^{3n} \times {(5)}^{6p} } \\ \\ [/tex]
[tex]\sf \: = \: {2}^{9m - 8m} \times {3}^{6n - 3n} \\ \\ [/tex]
[tex]\qquad\boxed{ \bf{ \: \because \: {x}^{m} \div {x}^{n} \: = \: {x}^{m - n} \: }} \\ \\ [/tex]
[tex]\sf \: = \: {2}^{m} \times {3}^{3n} \\ \\ [/tex]
Hence,
[tex]\sf\implies \bf \: \dfrac{ {8}^{3m} \times {9}^{3n} \times {(125)}^{2p} }{ {(16)}^{2m} \times {(27)}^{n} \times {(25)}^{3p} } = {2}^{m} \times {3}^{3n} \\ \\ [/tex]
[tex] \\ \large\underline{\sf{Solution-2}}[/tex]
Given that,
[tex]\sf \: x = 5 + 2 \sqrt{3} \\ \\ [/tex]
and
[tex]\sf \: y = 5 - 2 \sqrt{3} \\ \\ [/tex]
Now, Consider
[tex]\sf \: {7x}^{2} + 14xy + {7y}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 7( {x}^{2} + 2xy + {y}^{2}) \\ \\ [/tex]
[tex]\sf \: = \: 7 {(x + y)}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 7 {(5 + 2 \sqrt{3} + 5 - 2 \sqrt{3} )}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 7 {(10)}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 7 \times 100 \\ \\ [/tex]
[tex]\sf \: = \: 700 \\ \\ [/tex]
[tex]\sf\implies \bf \: {7x}^{2} + 14xy + {7y}^{2} = 700 \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{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]
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Verified answer
Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \:(1) \: \: \dfrac{ {8}^{3m} \times {9}^{3n} \times {(125)}^{2p} }{ {(16)}^{2m} \times {(27)}^{n} \times {(25)}^{3p} } = {2}^{m} \times {3}^{3n}\qquad \: \\ \\& \qquad \:\sf \: (2) \: \: {7x}^{2} + 14xy + {7y}^{2} = 700\end{aligned}} \qquad \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given expression is
[tex]\sf \: \dfrac{ {8}^{3m} \times {9}^{3n} \times {(125)}^{2p} }{ {(16)}^{2m} \times {(27)}^{n} \times {(25)}^{3p} } \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \dfrac{ {( {2}^{3}) }^{3m} \times {( {3}^{2}) }^{3n} \times {( {5}^{3} )}^{2p} }{ {( {2}^{4} )}^{2m} \times {( {3}^{3} )}^{n} \times {( {5}^{2} )}^{3p} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{ {2}^{9m} \times {3}^{6n} \times {(5)}^{6p} }{ {(2)}^{8m} \times {(3)}^{3n} \times {(5)}^{6p} } \\ \\ [/tex]
[tex]\sf \: = \: {2}^{9m - 8m} \times {3}^{6n - 3n} \\ \\ [/tex]
[tex]\qquad\boxed{ \bf{ \: \because \: {x}^{m} \div {x}^{n} \: = \: {x}^{m - n} \: }} \\ \\ [/tex]
[tex]\sf \: = \: {2}^{m} \times {3}^{3n} \\ \\ [/tex]
Hence,
[tex]\sf\implies \bf \: \dfrac{ {8}^{3m} \times {9}^{3n} \times {(125)}^{2p} }{ {(16)}^{2m} \times {(27)}^{n} \times {(25)}^{3p} } = {2}^{m} \times {3}^{3n} \\ \\ [/tex]
[tex] \\ \large\underline{\sf{Solution-2}}[/tex]
Given that,
[tex]\sf \: x = 5 + 2 \sqrt{3} \\ \\ [/tex]
and
[tex]\sf \: y = 5 - 2 \sqrt{3} \\ \\ [/tex]
Now, Consider
[tex]\sf \: {7x}^{2} + 14xy + {7y}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 7( {x}^{2} + 2xy + {y}^{2}) \\ \\ [/tex]
[tex]\sf \: = \: 7 {(x + y)}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 7 {(5 + 2 \sqrt{3} + 5 - 2 \sqrt{3} )}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 7 {(10)}^{2} \\ \\ [/tex]
[tex]\sf \: = \: 7 \times 100 \\ \\ [/tex]
[tex]\sf \: = \: 700 \\ \\ [/tex]
Hence,
[tex]\sf\implies \bf \: {7x}^{2} + 14xy + {7y}^{2} = 700 \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{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]