Math, asked by deiineideilamlhangha, 7 months ago

Simplify and answer each of the following in

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Answered by Mihir1001

$\huge{\underline{\mathfrak{\textcolor{blue}{Answer :}}}}$

$\huge\boxed{\fcolorbox{red}{pink}{answer}}$

$\huge{\underline{\mathrm{\textcolor{red}{Step-by-step \: \: explanation :}}}}$

$\LARGE{\underline{\mathtt{\textcolor{violet}{Given :-}}}}$

( a ) $\Large{ { \left( \frac{ - 3}{31} \right) }^{5} \times { \left( \frac{ - 3}{31} \right) }^{4} \div { \left( \frac{ - 3}{31} \right) }^{8} }$

( b ) $\Large{ { \left[ {( - 3)}^{2} \right] }^{3} \times \frac{1}{ { \left[ {( - 3)}^{3} \right] }^{2} } }$

( c ) $\Large{ \left( {5}^{17} \times {5}^{19} \right) \div {5}^{37} }$

( d ) $\Large{ \left[ { \left( \frac{ - 2}{5} \right) }^{7} \div { \left( \frac{ - 2}{5} \right) }^{3} \right] \times { \left( \frac{ - 2}{5} \right) }^{2} }$

$\LARGE{\underline{\mathtt{\textcolor{green}{To \: \: find :-}}}}$

Simplify all ( a ) , ( b ) , ( c ) and ( d ) .

$\LARGE{\underline{\mathtt{\textcolor{teal}{Concept \: \: used :-}}}}$

Simplification, and
Exponentiation

$\LARGE{\underline{\mathtt{\textcolor{blue}{Solution :-}}}}$

$\Large\boxed{a}$ $\Large{ { \left( \frac{ - 3}{31} \right) }^{5} \times { \left( \frac{ - 3}{31} \right) }^{4} \div { \left( \frac{ - 3}{31} \right) }^{8} }$

$\Large{ = { \left( \frac{ - 3}{31} \right) }^{5 + 4 - 8} }$

$\Large{ = { \left( \frac{ - 3}{31} \right) }^{10 - 8} }$

$\Large{ = { \left( \frac{ - 3}{31} \right) }^{2} }$

$\Large{ = \frac{ {( - 3)}^{2} }{ {(31)}^{2} } }$

$\Large{ = \frac{9}{961} }$

$\Large\boxed{b}$ $\Large{ { \left{ {( - 3)}^{2} \right} }^{3} \times \frac{1}{ { \left{ { {( - 3)}^{3} \right} }^{2} } } }$

$\Large{ = {( - 3)}^{2 \times 3} \times \frac{1}{ {( - 3)}^{3 \times 2} } }$

$\Large{ = {( - 3)}^{6} \times \frac{1}{ {( - 3)}^{6} } }$

$\Large{ = \cancel{ {( - 3)}^{6} } \times \frac{1}{ \cancel{ {( - 3)}^{6} } } }$

$\Large{ = 1 \times \frac{1}{1} }$

$\Large{ = 1}$

$\Large\boxed{c}$ $\Large{ \left( {5}^{17} \times {5}^{19} \right) \div {5}^{37} }$

$\Large{ = \left( {5}^{17 + 19} \right) \div {5}^{37} }$

$\Large{ = \left( {5}^{36} \right) \div {5}^{37} }$

$\Large{ = {5}^{36} \div {5}^{37} }$

$\Large{ = {5}^{36 - 37} }$

$\Large{ = {5}^{ - 1} }$

$\Large{ = \frac{1}{5} }$

$\Large{ = 0.2}$

$\Large\boxed{d}$ $\Large{ \left[ { \left( \frac{ - 2}{5} \right) }^{7} \div { \left( \frac{ - 2}{5} \right) }^{3} \right] \times { \left( \frac{ - 2}{5} \right) }^{2} }$

$\Large{ = \left{ { \left( \frac{ - 2}{5} \right) }^{7 - 3} \right} \times { \left( \frac{ - 2}{5} \right) }^{2} }$

$\Large{ = \left{ { \left( \frac{ - 2}{5} \right) }^{4} \right} \times { \left( \frac{ - 2}{5} \right) }^{2} }$

$\Large{ = { \left( \frac{ - 2}{5} \right) }^{4} \times { \left( \frac{ - 2}{5} \right) }^{2} }$

$\Large{ = { \left( \frac{ - 2}{5} \right) }^{4 + 2} }$

$\Large{ = { \left( \frac{ - 2}{5} \right) }^{6} }$

$\Large{ = \frac{ {( - 2)}^{6} }{ {(5)}^{6} } }$

$\Large{ = \frac{64}{15625} }$

$\LARGE{\underline{\mathtt{\textcolor{magenta}{Conclusion :-}}}}$

$\Large{( \: a \: ) \: \frac{9}{961} }$

$\Large{( \: b \: ) \: 1}$

$\Large{( \: c \: ) \: 0.2}$

$\Large{( \: d \: ) \: \frac{64}{15625} }$

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