Question

ch 12 exponents and power
answer the 8th question only
correct answer will be marked as brainlist immediately ​

Answer 1

$\large\underline{\sf{Given \:Question - 8(a)}}$

Simplify :-

$\rm \: \dfrac{ {10}^{ - 5} \times 125 \times {3}^{ - 5} }{ {6}^{ - 5} \times {5}^{7} }$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: \dfrac{ {10}^{ - 5} \times 125 \times {3}^{ - 5} }{ {6}^{ - 5} \times {5}^{7} }$

can be rewritten as

$\rm \: = \: \dfrac{ {(2 \times 5)}^{ - 5} \times (5 \times 5 \times 5) \times {3}^{ - 5} }{ {(3 \times 2)}^{ - 5} \times {5}^{7} }$

We know,

$\boxed{\sf{ \: {(xy)}^{n} = {x}^{n} \times {y}^{n} \: }}$

So, using this identity, we get

$\rm \: = \: \dfrac{ \cancel{ {2}^{ - 5}} \times \cancel{ {3}^{ - 5}} \times {5}^{3} \times {5}^{ - 5} }{ \cancel{{3}^{ - 5}} \times \cancel{ {2}^{ - 5}} \times {5}^{7} }$

$\rm \: = \: \dfrac{ {5}^{3} \times {5}^{ - 5} }{{5}^{7} }$

We know,

$\boxed{\sf{ \: {x}^{m} \div {x}^{n} \: = \: {x}^{m - n} \: }}$

So, using this identity, we get

$\rm \: = \: {5}^{3 - 7} \times {5}^{ - 5}$

$\rm \: = \: {5}^{- 4} \times {5}^{ - 5}$

$\rm \: = \: \dfrac{1}{ {5}^{4} \times {5}^{5} }$

$\rm \: = \: \dfrac{1}{{5}^{(4+5)} }$

$\rm \: = \: \dfrac{1}{{5}^{9} }$

$\rm \: = \: \dfrac{1}{1953125}$

Hence,

$\rm\implies \:\boxed{\sf{ \:\rm \: \dfrac{ {10}^{ - 5} \times 125 \times {3}^{ - 5} }{ {6}^{ - 5} \times {5}^{7} } = \frac{1}{1953125} }}$

$\large\underline{\bold{Given \:Question - 8(b)}}$

Simplify :-

$\rm \: \dfrac{ {5}^{4} \times {x}^{10} {y}^{5} }{ {5}^{4} \times {x}^{7} {y}^{4} }$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: \dfrac{ {5}^{4} \times {x}^{10} {y}^{5} }{ {5}^{4} \times {x}^{7} {y}^{4} }$

$\rm \: = \: \dfrac{\cancel{ {5}^{4}} \times {x}^{10} {y}^{5} }{ \cancel{{5}^{4}} \times {x}^{7} {y}^{4} }$

$\rm \: = \: {x}^{10 - 7} \times {y}^{5 - 4}$

$\rm \: = \: {x}^{3} \times y$

$\rm \: = \: {x}^{3} y$

Hence,

$\rm\implies \:\boxed{\sf{ \:\rm \: \dfrac{ {5}^{4} \times {x}^{10} {y}^{5} }{ {5}^{4} \times {x}^{7} {y}^{4} } = {x}^{3} y \: \: }}$

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Additional Information:-

$\boxed{\sf{ \: {x}^{m} \times {x}^{n} = {x}^{ m+ n} \: \: }}$

$\boxed{\sf{ \: {x}^{0} = 1 \: }}$

$\boxed{\sf{ \: {x}^{ - n} = \frac{1}{ {x}^{n} } \: }}$

$\boxed{\sf{ \: {\bigg(\dfrac{x}{y} \bigg) }^{ - n} = {\bigg(\dfrac{y}{x} \bigg) }^{n} \: }}$

Answer 2

Question:-

Simplify:

$\sf\large a) \: \frac{10 {}^{ - 5} \times 125 \times 3 {}^{ - 5} }{6 {}^{ - 5} \times 5 {}^{ - 7} } . \\ \\ \sf \large b) \: \: \frac{5 {}^{4} \times x {}^{10}y {}^{5} }{5 {}^{4} \times x{}^{7} y {}^{4} } .$

Given:-

$\sf\large a) \: \frac{10 {}^{ - 5} \times 125 \times 3 {}^{ - 5} }{6 {}^{ - 5} \times 5 {}^{ - 7} } . \\ \\ \sf \large b) \: \: \frac{5 {}^{4} \times x {}^{10}y {}^{5} }{5 {}^{4} \times x{}^{7} y {}^{4} } .$

Solution:-

$\sf\large a) \: \frac{10 {}^{ - 5} \times 125 \times 3 {}^{ - 5} }{6 {}^{ - 5} \times 5 {}^{ - 7} } . </p><p>$

$\sf \large = \frac{(3) {}^{ - 5} \times (10 ) {}^{ - 5} \times 125 }{(5) {}^{ - 7} \times (6) {}^{ - 5} }$

We know that,

$\sf \large (a {}^{m} \times a {}^{n} = a {}^{m + n} \: \: and \: \: a {}^{m} \div a {}^{n} = a {m - n}^{2} ).$

$\sf \large = \frac{(3) {}^{ - 5} \times (2 \times 5) {}^{ - 5} \times 5 {}^{3} }{(5) {}^{ - 7} \times (3 \times 2) {}^{ - 5} } = \frac{(3) {}^{ - 5} \times (2) {}^{ - 5} \times (5) {}^{ - 5} \times 5 {}^{3} }{5 {}^{ - 7} \times 2 {}^{ - 5} \times 3 {}^{ - 5} }$

$\sf \large = \frac{(5) {}^{ - 5} \times 5 {}^{3} }{5 {}^{ - 7} } = \frac{(5) {}^{ - 2} }{5 {}^{ - 7} } = 5 {}^{5}$

$\sf \large 5 {}^{5} = 3125.$

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$\sf \large b) \: \: \frac{5 {}^{4} \times x {}^{10}y {}^{5} }{5 {}^{4} \times x{}^{7} y {}^{4} } .$

$\sf \large \frac{5 {}^{4} \times \: \: x {}^{10} y {}^{2} }{5 {}^{4} \times \: \: x {}^{7} y {}^{4} } = 5 {}^{4 - 4} \times \: \: x {}^{10 - 7} y {}^{5 - 4}$

$\sf \large = 5 {}^{0} \times \: \: x {}^{3} y {}^{1}$

$\sf \large = 1 \times \: \: x{}^{3} y$

$\sf \large = x {}^{3} y.$

Answer:-

$\sf \large a) \: \frac{10 {}^{ - 5} \times 125 \times 3 {}^{ - 5} }{6 {}^{ - 5} \times 5 {}^{ - 7} } = 3125.\\ \\ \sf \large b) \: \: \frac{5 {}^{4} \times x {}^{10}y {}^{5} }{5 {}^{4} \times x{}^{7} y {}^{4} } = x {}^{3} y.$

Hope you have satisfied. ⚘