Question

simplify. aa fast as possible​

Answer 1

Solution (1) :

8 = 2×2×2 = 2³

${8}^{x} = { ({2}^{3}) }^{x} = {2}^{3x}$

${16}^{x + 3} = {( {2}^{4} )}^{x + 3} = {2}^{4x + 12}$

Solution (2) :

Given that :

$a = {2}^{m} \: and \: b = {2}^{m + 1}$

To show :

$\frac{8 {a}^{3} }{ {b}^{2} } = {2}^{m + 1}$

On taking L.H.S. :

$\frac{8 {a}^{3} }{ {b}^{2} } \\ \\ = > \frac{8 \times {( {2}^{m}) }^{3} }{ {( {2}^{m + 1}) }^{2} } = \frac{8 \times {2}^{3m} }{ {2}^{2m + 2} } \\ \\ = > \frac{ {2}^{3} \times {2}^{3m} }{ {2}^{2m + 2} } \\ \\ = > \frac{ {2}^{3 + 3m} }{ {2}^{2m + 2} } \\ \\ = > {2}^{3 + 3m - 2m - 2} \\ \\ = > {2}^{m + 1} = rhs$

HENCE PROVED ✔️

Solution (3) :

Given that :

$x = {2}^{k} \: and \: y = {2}^{k + 3}$

Then,

$\frac{x}{y} = \frac{ {2}^{k} }{ {2}^{k + 3} } \\ \\ = > \frac{x}{y} = {2}^{k - k - 3} \\ \\ = > \frac{x}{y} = {2}^{ - 3} = \frac{1}{ {2}^{3} } \\ \\ = > \frac{x}{y} = \frac{1}{8}$

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