Question

Can anyone help me out with this?​

Answer 1

${\large{\textsf{\textbf{\underline{\underline{Question \: 1 :}}}}}}$

$\star\:{\underline{\underline{\sf{\purple{Solution:}}}}}$

$\bullet \sf \: {(a + b)}^{ab}$

Putting value of a as 3 and b as -2, we get :

$\longrightarrow \sf \: {( 3 + (- 2))}^{3 \times - 2}$

$\longrightarrow \sf \: {( 3 - 2)}^{3 \times - 2}$

$\longrightarrow \sf \: {( 1)}^{ - 6}$

• Using negative Exponents Law

$\longrightarrow \sf \dfrac{1}{ {1}^{6} }$

$\longrightarrow \sf \dfrac{1}{ 1 \times 1 \times 1 \times 1 \times 1 \times 1 }$

$\longrightarrow \sf \dfrac{1}{ 1 }$

$\longrightarrow \sf \purple{1}$

${\large{\textsf{\textbf{\underline{\underline{Question \: 2 :}}}}}}$

$\star\:{\underline{\underline{\sf{\red{Solution:}}}}}$

$\bullet \sf \: \dfrac{ {8}^{ - 1} \times {5}^{3} }{ {2}^{ - 4}}$

$\longrightarrow \sf \: {8}^{ - 1} \times {5}^{3} \times \dfrac{1}{{2}^{ - 4}}$

• Using negative Exponents Law

$\longrightarrow \sf \: {8}^{ - 1} \times {5}^{3} \times {2}^{4}$

$\longrightarrow \sf \: {8}^{ - 1} \times 5 \times 5 \times 5 \times {2}^{4}$

$\longrightarrow \sf \: {8}^{ - 1} \times 125 \times {2}^{4}$

$\longrightarrow \sf \: {8}^{ - 1} \times 125 \times 2 \times 2 \times 2 \times 2$

• Using negative Exponents Law

$\longrightarrow \sf \: \dfrac{1}{ \cancel{8}_{4}} \times 125 \times \cancel{2}_{1} \times 2 \times 2 \times 2$

$\longrightarrow \sf \: \dfrac{1}{ \cancel4_{2}} \times 125 \times \cancel{2}_{1} \times 2 \times 2$

$\longrightarrow \sf \: \dfrac{1}{ \cancel2} \times 125 \times \cancel{2} \times 2$

$\longrightarrow \sf \: 125 \times 2$

$\longrightarrow \sf \red{ 250}$

${\large{\textsf{\textbf{\underline{\underline{Question \: 3 :}}}}}}$

$\star\:{\underline{\underline{\sf{\green{Solution(1):}}}}}$

$\bullet \sf \dfrac{ \sqrt{32} + \sqrt{48} }{ \sqrt{8} + \sqrt{12} }$

$\longrightarrow \sf \dfrac{ \sqrt{4 \times 4 \times 2} + \sqrt{4 \times 4 \times 3} }{ \sqrt{2 \times 2 \times 2} + \sqrt{2 \times 2 \times 3} }$

$\longrightarrow \sf \dfrac{ \sqrt{ {4}^{2} \times 2} + \sqrt{ {4}^{2} \times 3} }{ \sqrt{ {2}^{2} \times 2} + \sqrt{ {2}^{2} \times 3} }$

$\longrightarrow \sf \dfrac{ 4\sqrt{ 2} + 4 \sqrt{ 3} }{ 2\sqrt{ 2} +2 \sqrt{ 3} }$

$\longrightarrow \sf \dfrac{ \cancel{ 4}_{2}(\sqrt{ 2} + \sqrt{ 3}) }{ \cancel{2}(\sqrt{ 2} + \sqrt{ 3}) }$

$\longrightarrow \sf \dfrac{ 2 \: \cancel{(\sqrt{ 2} + \sqrt{ 3}) } }{ \cancel{(\sqrt{ 2} + \sqrt{ 3})} }$

$\longrightarrow \sf \green{2}$

$\star\:{\underline{\underline{\sf{\blue{Solution(2):}}}}}$

$\bullet \sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{80} + \sqrt{48} - \sqrt{45} - \sqrt{27} }$

$\begin{gathered} \longrightarrow \sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{4 \times 4 \times 5} + \sqrt{4 \times 4 \times 3} - \sqrt{3 \times 3 \times 5} - \sqrt{3 \times 3 \times 3} } \end{gathered}$

$\begin{gathered}\longrightarrow \sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{ {4}^{2} \times 5} + \sqrt{ {4}^{2} \times 3} - \sqrt{ {3}^{2} \times 5} - \sqrt{ {3}^{2} \times 3} } \end{gathered}$

$\longrightarrow \sf \dfrac{ \sqrt{5} + \sqrt{3} }{4 \sqrt{ 5} + 4 \sqrt{ 3} - 3\sqrt{ 5} - 3\sqrt{ 3} }$

$\longrightarrow \sf \dfrac{ \sqrt{5} + \sqrt{3} }{4 \sqrt{ 5} - 3\sqrt{ 5} + 4 \sqrt{ 3} - 3\sqrt{ 3} }$

$\longrightarrow \sf \dfrac{ \cancel{ \sqrt{5} + \sqrt{3}} }{ \cancel{\sqrt{ 5} + \sqrt{ 3} } }$

$\longrightarrow \blue{1}$

${\large{\textsf{\textbf{\underline{\underline{Answers :}}}}}}$

• Question 1 - $\purple{1}$

• Question 2 - $\red{250}$

• Question 3(1) - $\green{2}$

• Question 3(2) - $\blue{1}$

${\large{\textsf{\textbf{\underline{\underline{ Concept \: :}}}}}}$

★ Negative Exponents Law -

$\bullet \sf \: {a}^{ - m} = \dfrac{1}{ {a}^{m} }$

★ $\sqrt{32}$ can be written as $4 \sqrt{2}$

‣ $\sqrt{48}$ can be written as $4 \sqrt{3}$

‣ $\sqrt{8}$ can be written as $2 \sqrt{2}$

‣ $\sqrt{12}$ can be written as $2 \sqrt{3}$

‣ $\sqrt{80}$ can be written as $4 \sqrt{5}$

‣ $\sqrt{48}$ can be written as $4 \sqrt{3}$

‣ $\sqrt{45}$ can be written as $3 \sqrt{5}$

‣ $\sqrt{27}$ can be written as $3 \sqrt{3}$

★ 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.

${\large{\textsf{\textbf{\underline{\underline{ Note \: :}}}}}}$

• Swipe to see the full answer.

$\begin{gathered} {\underline{\rule{330pt}{3pt}}} \end{gathered}$