Question

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Answer 1

Given Question

$\rm :\longmapsto\: \: {8}^{a} = {16}^{b + 2}$

$\rm :\longmapsto\: \: {27}^{b} = {9}^{a + 1}$

$\rm :\longmapsto\:a - b = - - -$

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Given that,

$\rm :\longmapsto\: \: {8}^{a} = {16}^{b + 2}$

$\rm :\longmapsto\: {( {2}^{3} )}^{a} = {( {2}^{4})}^{b + 2}$

$\rm :\longmapsto\: \: {2}^{3a} = {2}^{4(b + 2)}$

$\rm\implies \:3a = 4b + 8$

$\rm\implies \:3a - 4b = 8 - - - (1)$

Also, given that

$\rm :\longmapsto\: \: {27}^{b} = {9}^{a + 1}$

$\rm :\longmapsto\: \: {( {3}^{3} )}^{b} = {( {3}^{2} )}^{a + 1}$

$\rm :\longmapsto\: \: {3}^{3b} = {3}^{2(a + 1)}$

$\rm\implies \:3b = 2a + 2$

$\rm\implies \:3b - 2a = 2 - - - (2)$

On adding equation (1) and (2), we get

$\rm :\longmapsto\:3a - 4b + 3b - 2a = 8 + 2$

$\rm :\longmapsto\:a - b = 10$

Hence,

$\rm\implies \:\boxed{\tt{ a \: - \: b \: = \: 10 \: }}$

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Explore More

$\rm :\longmapsto\:\boxed{\tt{ {x}^{m} \times {x}^{n} = {x}^{m + n}}}$

$\rm :\longmapsto\:\boxed{\tt{ {x}^{m} \div {x}^{n} = {x}^{m - n}}}$

$\rm :\longmapsto\:\boxed{\tt{ {( {x}^{m} )}^{n} = {x}^{mn}}}$

$\rm :\longmapsto\:\boxed{\tt{ {x}^{0} = 1}}$

$\rm :\longmapsto\:\boxed{\tt{ {x}^{ - n} = \frac{1}{ {x}^{n} } }}$

Answer 2

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