Question

find n so that
${2}^{11} \div {2}^{5} = {2}^{ - 3} \times {2}^{2n - 1}$
​

Answer 1

Solution:-

$\sf{{2}^{11} \div {2}^{5} = {2}^{ - 3} \times {2}^{2n - 1}}$

$\sf \red{ {a}^{m} \div {a}^{n} = {a}^{m - n} }$

$\sf \red{ {a}^{m} \times {a}^{n} = {a}^{m + n} }$

$\sf{{2}^{11 - 5} = {2}^{ - 3 +2 n - 1}}$

$\sf{{2}^{6} = {2}^{ 2 n - 1 - 3 }}$

$\sf{{2}^{6} = {2}^{ 2 n - 4 }}$

$\sf \red{6 = 2n - 4}$

$\sf \red{6 + 4= 2n }$

$\sf \red{10= 2n }$

$\sf \red{ \frac{ \cancel10}{ \cancel2} = n }$

$\sf{n = 5}$

Answer 2

Question:-

$\frac{ {2}^{11} }{ {2}^{5} } = {2}^{ - 3} \times {2}^{2n - 1}$

Required Formulae:-

${a}^{m - n} = \frac{ {a}^{m} }{ {a}^{n} }$

${a}^{m} \times {a}^{n} = {a}^{m + n}$

$( {a}^{m} ) ^{n} = {a}^{mn}$

Solution:-

Using formulae question becomes

${2}^{11 - 5} = {2}^{( - 3) + 2n - 1}$

${2}^{6} = {2}^{2n - 4}$

Bases are equal. So, powers also equal

$6 = 2n - 4$

$2n = 6 + 4$

$2n = 10$

$n = \frac{10}{2} = 5$

Final Answer:-

The value of n is 5