Question

Challenge.. ☄️⚡

$\large \sf1. Find \: the \: valu \: of \: x\ \textless \ br /\ \textgreater \ \: \sf {8}^{2x + 2} = {16}^{3x - 1} \\ \\ \ \textless \ br /\ \textgreater \ \large \sf2. Evaluate :-\ \textless \ br /\ \textgreater \ \sf\frac{36^{ \frac{5}{2}} - 36^{ \frac{7}{2} } }{36^{ \frac{3}{2} } }$
​

Answer 1

Answer:

$\large\underline{\underline{\bigstar{\textbf{\textsf{\:Question.1\::-}}}}}$

$\quad{: \longmapsto {\sf {(8)}^{2x + 2} = {(16)}^{3x - 1}}}$

$\begin{gathered}\end{gathered}$

$\large\underline{\underline{\bigstar{\textbf{\textsf{\:Solution\::-}}}}}$

${: \longmapsto {\sf {(8)}^{2x + 2} = {(16)}^{3x - 1}}}$

${: \longmapsto {\sf {({2}^{3})}^{2x + 2} = {({2}^{4} )}^{3x - 1}}}$

${: \longmapsto {\sf {({2}^{})}^{(2x \times 3) +( 2 \times 3)} = {({2} )}^{(3x \times 4) - (1 \times 4)}}}$

${: \longmapsto {\sf {({2})}^{6x +6} = {({2} )}^{12x - 4}}}$

${: \longmapsto {\sf {( \cancel{2})}^{6x +6} = {( \cancel{2} )}^{12x - 4}}}$

${: \longmapsto {\sf{6x +6 ={12x - 4}}}}$

${: \longmapsto {\sf{6x - 12x = - 4 - 6}}}$

${: \longmapsto {\sf{ - 6x= - 10}}}$

${: \longmapsto {\sf{ \cancel{-} 6x= \cancel{-}10}}}$

${: \longmapsto {\sf{6x= 10}}}$

${: \longmapsto {\sf{x= \dfrac{10}{6}}}}$

${: \longmapsto {\sf{x= \cancel\dfrac{10}{6}}}}$

${: \longmapsto {\sf{x= \dfrac{5}{3}}}}$

${\bigstar{\underline{\boxed{\pmb{\sf{\red{x= \dfrac{5}{3}}}}}}}}$

∴ The value of x is 5/3.

$\begin{gathered}\end{gathered}$

$\large\underline{\underline{\bigstar{\textbf{\textsf{\:Verification\::-}}}}}$

${: \longmapsto {\sf {(8)}^{2x + 2} = {(16)}^{3x - 1}}}$

• Substuting the values of x = 5/3

${: \longmapsto {\sf {(8)}^{(2 \times \frac{5}{3} ) + 2} = {(16)}^{(3 \times \frac{5}{3}) - 1}}}$

${: \longmapsto {\sf {(8)}^{\frac{10}{3} + 2} = {(16)}^{\frac{15}{3} - 1}}}$

${: \longmapsto {\sf {(8)}^{\frac{10}{3} + \frac{2}{1} } = {(16)}^{\frac{15}{3} - \frac{1}{1} }}}$

${: \longmapsto {\sf {(8)}^{\frac{10 + 6}{3} } = {(16)}^{\frac{15 - 3}{3} }}}$

${: \longmapsto {\sf {(8)}^{\frac{16}{3} } = {(16)}^{\frac{12}{3} }}}$

${: \longmapsto {\sf {(8)}^{\frac{16}{3} } = {(16) \:}^ { \cancel\frac{12}{3} }}}$

${: \longmapsto {\sf {(8)}^{\frac{16}{3} } = {(16)}^{4 }}}$

${: \longmapsto {\sf {({2}^{3})}^{\frac{16}{3}} = {({2}^{4}) }^{4 }}}$

${: \longmapsto {\sf {({2})}^{3 \times \frac{16}{3}} = {({2}) }^{4 \times 4}}}$

${: \longmapsto {\sf {({2})}^{ \cancel{3} \times \frac{16}{\cancel{3}}} = {({2}) }^{4 \times 4}}}$

${: \longmapsto {\sf {{(2)}^{16} = {(2)}^{16}}}}$

${\bigstar{\underline{\boxed{\pmb{\sf{\red{LHS=RHS}}}}}}}$

∴ Hence Verified!!

━━━━━━━━━━━━━━━━━━━━

$\large\underline{\underline{\bigstar{\textbf{\textsf{\:Question.2\::-}}}}}$

${: \implies{\sf\dfrac{{(36)}^{ \frac{5}{2}} - {(36)}^{ \frac{7}{2} } }{{(36)}^{ \frac{3}{2} } }}}$

$\begin{gathered}\end{gathered}$

$\large\underline{\underline{\bigstar{\textbf{\textsf{\:Solution\::-}}}}}$

${: \implies{\sf\dfrac{{(36)}^{ \frac{5}{2}} - {(36)}^{ \frac{7}{2} } }{{(36)}^{ \frac{3}{2} } }}}$

${: \implies{\sf\dfrac{{( {6}^{2} )}^{ \frac{5}{2}} - {( {6}^{2} )}^{ \frac{7}{2} } }{{( {6}^{2} )}^{ \frac{3}{2} } }}}$

${: \implies{\sf\dfrac{{( {6} )}^{2 \times \frac{5}{2}} - {( {6})}^{ 2 \times \frac{7}{2} } }{{( {6} )}^{2 \times \frac{3}{2} } }}}$

${: \implies{\sf\dfrac{{( {6} )}^{ \cancel{2} \times \frac{5}{\cancel{2}}} - {( {6})}^{ \cancel{2} \times \frac{7}{\cancel{2}}} }{{( {6} )}^{\cancel{2} \times \frac{3}{\cancel{2} }} }}}$

${: \implies{\sf\dfrac{{( {6} )}^{{5}} - {( {6})}^{ {7} } }{{( {6} )}^{{3} } }}}$

${: \implies{\sf\dfrac{{(6 \times 6 \times 6 \times 6 \times 6)} - {( 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6)}}{{{(6 \times 6 \times 6)}}}}}$

${: \implies{\sf\dfrac{ 7776 - 279936}{ 216}}}$

${: \implies{\sf\dfrac{-272160}{ 216}}}$

${:\implies{\sf{ \cancel{\dfrac{-272160}{ 216}}}}}$

${: \implies{\sf{-1260}}}$

$\bigstar{\underline{\boxed{\pmb{\sf{\red{-1260}}}}}}$

∴ The required answer is -1260.

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Answer 2

Step-by-step explanation:

$\large\textsf{Question 1:}$

⇒ $(8) ^{2x + 2} = (16) ^{3x - 1}$

$\large\textsf{Answer:}$

⇒ $(8) ^{2x + 2} = (16) ^{3x - 1}$

⇒ $(2³)^{2x + 2} = (2⁴)^{3x - 1}$

⇒ $(2)^{(2x \times 3) + (2 \times 3)} = (2)^{(3x \times 4) - (1 \times 4)}$

⇒ $(2) ^{6x + 6} = (2)^{12x - 4}$

⇒ 6x+6 = 12x-4

⇒ 6x - 12x = -4 - 6

⇒ -6x = -10

⇒ 6x = 10

⇒ x = $\frac{10}{6}$

⇒ x = $\frac{5}{3}$

∴ The value of x is $\frac{5}{3}$.

$\large\textsf{Question 2:}$

⇒ $\large\frac{(36) ^{ \frac{3}{2}} - (36) ^{ \frac{7}{2} } }{(36) ^{ \frac{3}{2} } }$

$\large\textsf{Answer:}$

⇒ $\large\frac{(36) ^{ \frac{3}{2}} - (36) ^{ \frac{7}{2} } }{(36) ^{ \frac{3}{2} } }$

⇒ $\large\frac{(6²) ^{ \frac{3}{2}} - (6²) ^{ \frac{7}{2} } }{(6²) ^{ \frac{3}{2} } }$

⇒ $\large\frac{(6) ^{2 \times \frac{5}{2} } - (6) ^{2 \times \frac{7}{2} } }{(6) ^{2 \times \frac{3}{2} } }$

⇒ $\large\frac{(6) ^{5} - \: (6) ^{7} }{(6) ^{3} }$

⇒ $\large\frac{(6 \times 6 \times 6 \times 6 \times 6 \times 6) - (6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6) }{(6 \times 6 \times 6 )}$

⇒ $\large\frac{7776 - 279936}{216}$

⇒ $\large \frac{ - 272160}{216}$

⇒ -1260

∴ The required answer is -1260.