[tex]\star\large{ \underline{ \underline{ \boxed{ \purple{ \pmb{ \mathfrak{Question:-}}}}}}}[/tex]
[tex]\begin{gathered}\begin{gathered} \\ { \qquad{ \rule{280pt}{2.5pt}}} \end{gathered}\end{gathered}[/tex]
Simplify the following:
[tex]\begin{gathered}\begin{gathered} \\ \sf \: \bull \: \: \frac{( {7}^{n + 2}) - \times ({6 \times 7}^{n + 1})}{(21 \times {7}^{n}) - (2 \times {7}^{n + 1}) } \\ \\ \end{gathered}\end{gathered} [/tex]
[tex]\begin{gathered}\begin{gathered} \\ \sf \: \bull \: \: \frac{3^{2(2n + 6)} \times {(9)}^{2n - 7} } { {(27)}^{2n} } \\ \\ \end{gathered}\end{gathered} [/tex]
[tex]\begin{gathered}\begin{gathered} \\ \sf \: \bull \: \: (\frac{1}{9})^{ - 2} - 3 \times {27}^{ \frac{2}{3} } \times {9}^{0} + (\frac{9}{81})^{ - \frac{1}{2} } \\ \\ \end{gathered}\end{gathered} [/tex]
[tex]\begin{gathered}\begin{gathered} \\ \\ { \underline{ \rule{300pt}{8pt}}} \end{gathered}\end{gathered} [/tex]
[tex]\begin{gathered}\begin{gathered} \\ { \qquad{ \rule{280pt}{2.5pt}}} \end{gathered}\end{gathered}[/tex]
Simplify the following:
[tex]\begin{gathered}\begin{gathered} \\ \sf \: \bull \: \: \frac{( {7}^{n + 2}) - \times ({6 \times 7}^{n + 1})}{(21 \times {7}^{n}) - (2 \times {7}^{n + 1}) } \\ \\ \end{gathered}\end{gathered} [/tex]
[tex]\begin{gathered}\begin{gathered} \\ \sf \: \bull \: \: \frac{3^{2(2n + 6)} \times {(9)}^{2n - 7} } { {(27)}^{2n} } \\ \\ \end{gathered}\end{gathered} [/tex]
[tex]\begin{gathered}\begin{gathered} \\ \sf \: \bull \: \: (\frac{1}{9})^{ - 2} - 3 \times {27}^{ \frac{2}{3} } \times {9}^{0} + (\frac{9}{81})^{ - \frac{1}{2} } \\ \\ \end{gathered}\end{gathered} [/tex]
[tex]\begin{gathered}\begin{gathered} \\ \\ { \underline{ \rule{300pt}{8pt}}} \end{gathered}\end{gathered} [/tex]
Answers & Comments
Verified answer
Answer:
[ (7^(n+2) ) - (6 * 7^(n+1) ) ]/ [ (21*7^n) - (2*7^(n+1) ) ]
=7^(n+1) [7 - 6 ] / 7^n[ 21-14]
=7^(n+1-n) * (1/7)
=7*1/7
=1
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[tex]\displaystyle \sf{ \frac{ {(3)}^{2(n + 6)} \times {(9)}^{2n - 7} }{ {(27)}^{2n} } }[/tex]
[tex]\displaystyle \sf{ = \frac{ {(3)}^{2n + 12} \times {( {3}^{2} )}^{2n - 7} }{ {( {3}^{3} )}^{2n} } }[/tex]
[tex]\displaystyle \sf{ = \frac{ {(3)}^{2n + 12} \times {( 3 )}^{4n - 14} }{ {(3 )}^{6n} } }[/tex]
[tex]\displaystyle \sf{ = \frac{ {(3)}^{2n + 12 + 4n - 14}}{ {(3 )}^{6n} } }[/tex]
[tex]\displaystyle \sf{ = \frac{ {(3)}^{6n - 2}}{ {(3 )}^{6n} } }[/tex]
[tex]\displaystyle \sf{ = {(3)}^{6n - 2 - 6n}}[/tex]
[tex]\displaystyle \sf{ = {(3)}^{ - 2 }}[/tex]
[tex]\displaystyle \sf{ = \frac{1}{ {3}^{2} } }[/tex]
[tex]\displaystyle \sf{ = \frac{1}{ 9} }[/tex]
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[tex]\left(\dfrac{1}{9}\right)^{-2} - 3 \times 27^{\frac{2}{3}} \times 9^0 + \left(\dfrac{9}{81}\right)^{\frac{-1}{2}}[/tex]
[tex]= \left(9\right)^{2} - 3 \times (3^3)^{\frac{2}{3}} \times 9^0 + \left(\dfrac{81}{9}\right)^{\frac{1}{2}}[/tex]
[tex]= \left(9\right)^{2} - 3 \times (3)^{\frac{6}{3}} \times 9^0 +\left(\frac{9}{1}\right)^{\frac{1}{2}}[/tex]
[tex]= \left(9\right)^{2} - 3 \times (3)^{2} \times 9^0 + \left(9\right)^{\frac{1}{2}}[/tex]
[tex]= 81 - 3\times 9 \times 1 + 3[/tex]
[tex]= 84 - 27[/tex]
[tex]= 57[/tex]
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