Question

$\sf \: If 3x−y=12, what \: is \: the \: value \: of \: \frac{ {8}^{x} }{ {2}^{y} } \ \textless \ br /\ \textgreater \ \: ?$
A) 212
B) 44
C) 82
D) The value cannot be determined from the information given.


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Answer 1

$\textsf{\large{\underline{Solution}:}}$

Given Information:

$\rm \longrightarrow 3x - y = 12$

Now consider the expression:

$\rm = \dfrac{ {8}^{x} }{ {2}^{y} }$

Can be written as:

$\rm = \dfrac{( {2}^{3} )^{x} }{ {2}^{y} }$

$\rm = \dfrac{{2}^{3x}}{ {2}^{y} }$

$\rm ={2}^{3x - y}$

We know that 3x - y = 12. Therefore, we get:

$\rm ={2}^{12}$

$\rm =4096$

★ Which is our required answer.

$\textsf{\large{\underline{Learn More}:}}$

$\rm 1. \: \: {a}^{m} \times {a}^{n} = {a}^{m + n}$

$\rm 2. \: \: ({a}^{m})^{n} = {a}^{mn}$

$\rm 3. \: \: \dfrac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n}$

$\rm4. \: \: {a}^{m} \times {b}^{m} = {(ab)}^{m}$

$\rm5. \: \: \bigg(\dfrac{a}{b} \bigg)^{m} = \dfrac{ {a}^{m} }{ {b}^{m} }$

$\rm6. \: \: {a}^{ - n} = \dfrac{1}{ {a}^{n} }$

$\rm7. \: \: {a}^{n} = {b}^{n} \rightarrow a = b, n \neq0$

$\rm8. \: \: {a}^{m} = {a}^{n} \rightarrow m = n, a \neq 1$

Answer 2

Answer:

⟶3x−y=12

Now consider the expression:

\rm = \dfrac{ {8}^{x} }{ {2}^{y} }=2y8x

Can be written as:

\rm = \dfrac{( {2}^{3} )^{x} }{ {2}^{y} }=2y(23)x

\rm = \dfrac{{2}^{3x}}{ {2}^{y} }=2y23x

\rm ={2}^{3x - y}=23x−y

We know that 3x - y = 12. Therefore, we get:

\rm ={2}^{12}=212

\rm =4096=4096

★ Which is our required answer.

\textsf{\large{\underline{Learn More}:}}Learn More:

\rm 1. \: \: {a}^{m} \times {a}^{n} = {a}^{m + n}1.am×an=am+n

\rm 2. \: \: ({a}^{m})^{n} = {a}^{mn}2.(am)n=amn

\rm 3. \: \: \dfrac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n}3.anam=am−n

\rm4. \: \: {a}^{m} \times {b}^{m} = {(ab)}^{m}4.am×bm=(ab)m

\rm5. \: \: \bigg(\dfrac{a}{b} \bigg)^{m} = \dfrac{ {a}^{m} }{ {b}^{m} }5.(ba)m=bmam

\rm6. \: \: {a}^{ - n} = \dfrac{1}{ {a}^{n} }6.a−n=an1

\rm7. \: \: {a}^{n} = {b}^{n} \rightarrow a = b, n \neq0