Question

Simplify and express :-

In the attachment।​

Answer 1

EXPLANATION.

$\sf \implies (1) = \dfrac{(10)^{2} \ \times (15)^{3} }{(2)^{2} \ \times 3 \ \times (5)^{5} \ \times (6)^{4} }$

As we know that,

In this type of we can simplify or expand the equation, we get.

$\sf \implies \dfrac{10 \ \times 10 \times 15 \times 15 \times 15 }{2 \times 2 \times 3 \times 5 \times 5 \times 5 \times 5 \times 5 \times 6 \times 6 \times 6 \times 6}$

$\sf \implies \dfrac{10 \times 10 \times 15 \times 15 \times 15}{10 \times 10 \times 15 \times 5 \times 5 \times 6 \times 6 \times 6 \times 6}$

$\sf \implies \dfrac{3 \times 3}{6 \times 6 \times 6 \times 6}$

$\sf \implies \dfrac{1}{2 \times 2 \times 6 \times 6} = \dfrac{1}{144}$

$\sf \implies (1) = \dfrac{(10)^{2} \ \times (15)^{3} }{(2)^{2} \ \times 3 \ \times (5)^{5} \ \times (6)^{4} } = \dfrac{1}{144}$

$\sf \implies (2) = \dfrac{(3)^{5} \times 25 \times (10)^{5} }{(5)^{7} \times (6)^{5} }$

As we know that,

In this type of we can simplify or expand the equation, we get.

$\sf \implies \dfrac{3 \times 3 \times 3 \times 3 \times 3 \times 25 \times 10 \times 10 \times 10 \times 10 \times 10}{5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 6 \times 6 \times 6 \times 6 \times 6}$

$\sf \implies \dfrac{3 \times 3 \times 3 \times 3 \times 3 \times 25 \times 100000}{25 \times 30 \times 30 \times 30 \times 30 \times 30 }$

$\sf \implies \dfrac{100000}{10 \times 10 \times 10 \times 10 \times 10} = 1.$

$\sf \implies (2) = \dfrac{(3)^{5} \times 25 \times (10)^{5} }{(5)^{7} \times (6)^{5} } = 1.$

Answer 2

✰ Topic :- Exponents.

$\frak{ \maltese \: \: Given} \: \begin{cases} \star \: \: \tt { \underline{ \textbf{ \textsf{(i) }}\: \: \dfrac{10 {}^{2} \times 15 {}^{3} }{2 {}^{2} \times 3 \times 5 {}^{5} \times 6 {}^{4} } }}\: \: \bigstar \\ \\\star \: \: \tt \underline{ \textsf{ \textbf{(ii) \: \: }} \dfrac{3 {}^{5} \times 25 \times 10 {}^{5} }{5 {}^{7} \times 6 {}^{5} } } \: \: \: \: \bigstar\end{cases}$

❍ Need To Do : Simplify and Express each of the following as a rational number.

$\maltese \: \: \large \underline{ \textsf{ \textbf{Required \: Solution}}} \: :$

${ \large{\star \: \: {\underline{ \frak{Solution }} \: _{ \textsf {\textbf{(number (1))}}}}}}$

${ \large\star} \: \: \tt { { \textbf{ \textsf{(i) }}\: \: \dfrac{10 {}^{2} \times 15 {}^{3} }{2 {}^{2} \times 3 \times 5 {}^{5} \times 6 {}^{4} } }}$

$\dag\: \: \frak{ \: \underline {Firstly, \: expanding \: the \: equation \: : }}$

$: \implies \tt \: \frac{(2 \times 5) {}^{2} \times (3 \times 5) {}^{3} }{(2) {}^{2} \times3 \times (5) {}^{5} \times (2 \times 3) {}^{4} }$

${ : \implies \tt \dfrac{2 \times 2 \times 5 \times 5 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5}{2 \times 2 \times 3 \times 5 \times 5 \times 5 \times 5 \times 5 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3} }$

$\dag \: \: \frak{ \: \underline {Secondly, \: cancelling \: them \: : }}$

${ : \implies \tt \dfrac{ \cancel{2 \times 2} \times{ \cancel{ 5 \times 5}} \times\cancel{ 3 \times 3 \times 3} \times \cancel{5 \times 5 \times 5}}{\cancel{2 \times 2 }\times 3 \times { \cancel{5 \times 5 \times 5 \times 5 \times 5}} \times 2 \times 2 \times 2 \times 2 \times\cancel{ 3 \times 3 \times 3 }\times 3} }$

$\dag \: \: \frak{ \: \underline {Thirdly, \: multiplying \: the \: denominators \: :}}$

$: \implies \tt \: \frac{1}{2 \times 2 \times 2 \times 2 \times3 \times 3}$

$: \implies \tt \: \frac{1}{16 \times 9}$

$\underline{ \underline{ \boxed {: \longrightarrow {\bf \: \frac{1}{144} }}}} \: \: \bigstar$

${ \large{\star \: \: {\underline{ \frak{Solution }} \: _{ \textsf {\textbf{(number (2))}}}}}}$

${ \large \star} \: \: \tt { \textsf{ \textbf{(ii) \: \: }} \dfrac{3 {}^{5} \times 25 \times 10 {}^{5} }{5 {}^{7} \times 6 {}^{5} } }$

$\dag\: \: \frak{ \: \underline {Firstly, \: expanding \: the \: equation \: : }}$

$: \implies \tt \: \frac{(3 ){}^{5} \times 5 \times 5 \times (2 \times 5) {}^{5} }{(5) {}^{7} \times (2 \times 3) {}^{5} }$

$: \implies \tt \: \frac{3 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5 \times 5}{5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3}$

$\dag \: \: \frak{ \: \underline {Secondly, \: cancelling \: them \: : }}$

$: \implies \tt \: \frac{ \cancel{3 \times 3 \times 3 \times 3 \times 3} \times { \cancel{5 \times 5}} \times { \cancel{2 \times 2 \times 2 \times 2 \times 2}} \times{ \cancel{ 5 \times 5 \times 5 \times 5 \times 5}}}{{ \cancel{5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5}} \times{ \cancel{ 2 \times 2 \times 2 \times 2 \times 2 }}\times { \cancel{3 \times 3 \times 3 \times 3 \times 3}}}$

$\underline{ \underline{ \boxed {: \longrightarrow {\bf \: 1}}}} \: \: \bigstar$