[tex]{\large{\textsf{\textbf{\underline{\underline{Question \: 1 :}}}}}}[/tex]
[tex]\star\:{\underline{\underline{\sf{\purple{Solution:}}}}}[/tex]
[tex] \bullet \sf \: {(a + b)}^{ab} [/tex]
Putting value of a as 3 and b as -2, we get :
[tex] \longrightarrow \sf \: {( 3 + (- 2))}^{3 \times - 2} [/tex]
[tex]\longrightarrow \sf \: {( 3 - 2)}^{3 \times - 2} [/tex]
[tex]\longrightarrow \sf \: {( 1)}^{ - 6} [/tex]
• Using negative Exponents Law
[tex]\longrightarrow \sf \dfrac{1}{ {1}^{6} } [/tex]
[tex]\longrightarrow \sf \dfrac{1}{ 1 \times 1 \times 1 \times 1 \times 1 \times 1 } [/tex]
[tex]\longrightarrow \sf \dfrac{1}{ 1 } [/tex]
[tex]\longrightarrow \sf \purple{1}[/tex]
[tex]{\large{\textsf{\textbf{\underline{\underline{Question \: 2 :}}}}}}[/tex]
[tex]\star\:{\underline{\underline{\sf{\red{Solution:}}}}}[/tex]
[tex] \bullet \sf \: \dfrac{ {8}^{ - 1} \times {5}^{3} }{ {2}^{ - 4}} [/tex]
[tex] \longrightarrow \sf \: {8}^{ - 1} \times {5}^{3} \times \dfrac{1}{{2}^{ - 4}} [/tex]
• Using negative Exponents Law
[tex]\longrightarrow \sf \: {8}^{ - 1} \times {5}^{3} \times {2}^{4} [/tex]
[tex]\longrightarrow \sf \: {8}^{ - 1} \times 5 \times 5 \times 5 \times {2}^{4} [/tex]
[tex]\longrightarrow \sf \: {8}^{ - 1} \times 125 \times {2}^{4}[/tex]
[tex]\longrightarrow \sf \: {8}^{ - 1} \times 125 \times 2 \times 2 \times 2 \times 2[/tex]
• Using negative Exponents Law
[tex]\longrightarrow \sf \: \dfrac{1}{ \cancel{8}_{4}} \times 125 \times \cancel{2}_{1} \times 2 \times 2 \times 2[/tex]
[tex]\longrightarrow \sf \: \dfrac{1}{ \cancel4_{2}} \times 125 \times \cancel{2}_{1} \times 2 \times 2[/tex]
[tex]\longrightarrow \sf \: \dfrac{1}{ \cancel2} \times 125 \times \cancel{2} \times 2[/tex]
[tex]\longrightarrow \sf \: 125 \times 2[/tex]
[tex]\longrightarrow \sf \red{ 250}[/tex]
[tex]{\large{\textsf{\textbf{\underline{\underline{Question \: 3 :}}}}}}[/tex]
[tex]\star\:{\underline{\underline{\sf{\green{Solution(1):}}}}}[/tex]
[tex] \bullet \sf \dfrac{ \sqrt{32} + \sqrt{48} }{ \sqrt{8} + \sqrt{12} } [/tex]
[tex] \longrightarrow \sf \dfrac{ \sqrt{4 \times 4 \times 2} + \sqrt{4 \times 4 \times 3} }{ \sqrt{2 \times 2 \times 2} + \sqrt{2 \times 2 \times 3} } [/tex]
[tex] \longrightarrow \sf \dfrac{ \sqrt{ {4}^{2} \times 2} + \sqrt{ {4}^{2} \times 3} }{ \sqrt{ {2}^{2} \times 2} + \sqrt{ {2}^{2} \times 3} } [/tex]
[tex] \longrightarrow \sf \dfrac{ 4\sqrt{ 2} + 4 \sqrt{ 3} }{ 2\sqrt{ 2} +2 \sqrt{ 3} } [/tex]
[tex]\longrightarrow \sf \dfrac{ \cancel{ 4}_{2}(\sqrt{ 2} + \sqrt{ 3}) }{ \cancel{2}(\sqrt{ 2} + \sqrt{ 3}) } [/tex]
[tex]\longrightarrow \sf \dfrac{ 2 \: \cancel{(\sqrt{ 2} + \sqrt{ 3}) } }{ \cancel{(\sqrt{ 2} + \sqrt{ 3})} } [/tex]
[tex]\longrightarrow \sf \green{2}[/tex]
[tex]\star\:{\underline{\underline{\sf{\blue{Solution(2):}}}}}[/tex]
[tex] \bullet \sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{80} + \sqrt{48} - \sqrt{45} - \sqrt{27} } [/tex]
[tex]\begin{gathered} \longrightarrow \sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{4 \times 4 \times 5} + \sqrt{4 \times 4 \times 3} - \sqrt{3 \times 3 \times 5} - \sqrt{3 \times 3 \times 3} } \end{gathered} [/tex]
[tex]\begin{gathered}\longrightarrow \sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{ {4}^{2} \times 5} + \sqrt{ {4}^{2} \times 3} - \sqrt{ {3}^{2} \times 5} - \sqrt{ {3}^{2} \times 3} } \end{gathered} [/tex]
[tex]\longrightarrow \sf \dfrac{ \sqrt{5} + \sqrt{3} }{4 \sqrt{ 5} + 4 \sqrt{ 3} - 3\sqrt{ 5} - 3\sqrt{ 3} } [/tex]
[tex]\longrightarrow \sf \dfrac{ \sqrt{5} + \sqrt{3} }{4 \sqrt{ 5} - 3\sqrt{ 5} + 4 \sqrt{ 3} - 3\sqrt{ 3} } [/tex]
[tex]\longrightarrow \sf \dfrac{ \cancel{ \sqrt{5} + \sqrt{3}} }{ \cancel{\sqrt{ 5} + \sqrt{ 3} } }[/tex]
[tex]\longrightarrow \blue{1}[/tex]
[tex]{\large{\textsf{\textbf{\underline{\underline{Answers :}}}}}}[/tex]
• Question 1 - [tex]\purple{1}[/tex]
• Question 2 - [tex]\red{250}[/tex]
• Question 3(1) - [tex]\green{2}[/tex]
• Question 3(2) - [tex]\blue{1}[/tex]
[tex]{\large{\textsf{\textbf{\underline{\underline{ Concept \: :}}}}}}[/tex]
★ Negative Exponents Law -
[tex] \bullet \sf \: {a}^{ - m} = \dfrac{1}{ {a}^{m} } [/tex]
★ [tex] \sqrt{32} [/tex] can be written as [tex]4 \sqrt{2} [/tex]
‣ [tex] \sqrt{48} [/tex] can be written as [tex]4 \sqrt{3} [/tex]
‣ [tex] \sqrt{8} [/tex] can be written as [tex]2 \sqrt{2} [/tex]
‣ [tex] \sqrt{12} [/tex] can be written as [tex]2 \sqrt{3} [/tex]
‣ [tex] \sqrt{80} [/tex] can be written as [tex]4 \sqrt{5} [/tex]
‣ [tex] \sqrt{48} [/tex] can be written as [tex]4 \sqrt{3} [/tex]
‣ [tex] \sqrt{45} [/tex] can be written as [tex]3 \sqrt{5} [/tex]
‣ [tex] \sqrt{27} [/tex] can be written as [tex]3 \sqrt{3} [/tex]
★ During Addition and Subtraction
• minus (-) minus (-) gives plus (+)
• minus (-) plus (+) gives minus (-)
• plus (+) minus (-) gives minus (-)
• plus (+) plus (+) gives plus (+)
• Also the sign of the resultant term depends upon the sign of the largest number.
[tex]{\large{\textsf{\textbf{\underline{\underline{ Note \: :}}}}}}[/tex]
• Swipe to see the full answer.
[tex]\begin{gathered} {\underline{\rule{330pt}{3pt}}} \end{gathered}[/tex]
