Question

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Answer 1

m = 35

Step-by-step explanation:

Using laws of Indices:

$</u></strong></p><p><strong><u>[tex] {(ab)}^{2} = {a}^{2} .{b}^{2} \\ \\ </u></strong></p><p><strong><u>[tex] {(ab)}^{2} = {a}^{2} .{b}^{2} \\ \\ (a^{2})^{3} = a^{2.3} = a^{6} \\ \\ </u></strong></p><p></p><p><strong><u>[tex] {(ab)}^{2} = {a}^{2} .{b}^{2} \\ \\ (a^{2})^{3} = a^{2.3} = a^{6} \\ \\ a^{4}.a^{3} = a ^{4+3} = a ^{7}$

Solution:

${ (\frac{1}{4}) }^{18} \times {( \frac{1}{5} )}^{m} = \frac{1}{2 \times {10}^{35} } \\ \\ {( \frac{1}{ {2}^{2} }) }^{18} \times \frac{1}{ {5}^{m} } = \frac{1}{2 \times {(5 \times 2)}^{35} } \\ \\ \frac{1}{ {2}^{(2 \times 18)} } \times \frac{1}{ {5}^{m} } = \frac{1}{2 \times {5}^{35} \times {2}^{35} } \\ \\ \frac{1}{ {2}^{36} } \times \frac{1}{ {5}^{m} } = \frac{1}{ {2}^{1 + 35} \times {5}^{35} } \\ \\ \frac{1}{ {2}^{36} } \times \frac{1}{ {5}^{m} } = \frac{1}{ {2}^{36} } \times \frac{1}{ {5}^{35} } \\ \\ \frac{1}{ {5}^{m} } = \frac{1}{ {5}^{35} } \\ \\ m = 35$

Hence the required value of m is 35 .

Answer 2

${\sf{\underline{\underline{\pink{Solution}}}}}$

$\sf\purple{m = 35}$

$\sf\orange{Step-by-step \ explanation:}$

$\sf\green{Using \ laws \ of \ indices}$

$(ab)^{2} = a ^{2} \times b^{2}$

$(a ^{2} )^{3} = a ^{2 \times 3} = a ^{6}$

$a ^{4} \times a ^{3} = a ^{4 + 7} \: = a^{7}$

$\sf\green{Solution:}$

$\begin{gathered}{ (\frac{1}{4}) }^{18} \times {( \frac{1}{5} )}^{m} = \frac{1}{2 \times {10}^{35} } \\ \\ {( \frac{1}{ {2}^{2} }) }^{18} \times \frac{1}{ {5}^{m} } = \frac{1}{2 \times {(5 \times 2)}^{35} } \\ \\ \frac{1}{ {2}^{(2 \times 18)} } \times \frac{1}{ {5}^{m} } = \frac{1}{2 \times {5}^{35} \times {2}^{35} } \\ \\ \frac{1}{ {2}^{36} } \times \frac{1}{ {5}^{m} } = \frac{1}{ {2}^{1 + 35} \times {5}^{35} } \\ \\ \frac{1}{ {2}^{36} } \times \frac{1}{ {5}^{m} } = \frac{1}{ {2}^{36} } \times \frac{1}{ {5}^{35} } \\ \\ \frac{1}{ {5}^{m} } = \frac{1}{ {5}^{35} } \\ \\ m = 35\end{gathered}$

$\sf\pink{Hence \ the \ required \ value \ of \ m \ is \ 35 .}$