Question

$\huge{\textbf{\textsf{{☆ QUE}}{\purple{ST}}{\pink{ION ☆} \: {{}{:}}}}}$



If 3x - y = 12,

What is the value of
$\frac{ {8}^{x} }{{2}^{y} }$



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Answer 1

Given : $\qquad \sf \leadsto \:\: 3x - y = 12 \:\: \qquad$

Exigency To Find : The value of $\dfrac{8^x}{2^y}$ .

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

❍ Let's 3x - y = 12 as Equation. 1 .

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Finding Value of $\bf \dfrac{8^x}{2^y}$ :

$\qquad \dashrightarrow \:\: \sf \dfrac{8^x}{2^y} \:\:$

$\qquad \dashrightarrow \:\: \sf \dfrac{8^x}{2^y} \:\:$

$\dag\:\:\sf{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\bigg\lgroup \sf{ 2^3 \:\: or\:\:2 \times 2\times 2 \: =\:\:8 }\bigg\rgroup$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Applying \: this \::}}$

$\qquad \dashrightarrow \:\: \sf \dfrac{8^x}{2^y} \:\:$

$\qquad \dashrightarrow \:\: \sf \dfrac{(2^3)^x}{2^y} \:\:$

$\dag\:\:\sf{ As,\:We\:know\:that\::}\\\\ \qquad\maltese\bf \:\: Law's\:of\:Exponent\:\::$

$\qquad \dag\:\:\bigg\lgroup \sf{ (a^m)^n = a \:^{m \times n} }\bigg\rgroup$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Applying \: this \::}}$

$\qquad \dashrightarrow \:\: \sf \dfrac{(2^3)^x}{2^y} \:\:$

$\qquad \dashrightarrow \:\: \sf \dfrac{(2)^{3 \times x}}{2^y} \:\:$

$\qquad \dashrightarrow \:\: \sf \dfrac{(2)^{3x}}{2^y} \:\:$

$\qquad \dashrightarrow \:\: \sf \dfrac{2^{3x}}{2^y} \:\:$

$\dag\:\:\sf{ As,\:We\:know\:that\::}\\\\ \qquad\maltese\bf \:\: Law's\:of\:Exponent\:\::$

$\qquad \dag\:\:\bigg\lgroup \sf{ \dfrac{a^m}{a^n} = a \:^{m - n} }\bigg\rgroup$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Applying \: this \::}}$

$\qquad \dashrightarrow \:\: \sf \dfrac{2^{3x}}{2^y} \:\:$

$\qquad \dashrightarrow \:\: \sf 2^{3x- y} \:\:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: Equation \:\:1 \: Value\::}}$

$\qquad \dag\:\:\bigg\lgroup \sf{ Equation \:\:1 \:=\:3x - y = 12 }\bigg\rgroup$

$\qquad \dashrightarrow \:\: \sf 2^{12} \:\:$

$\qquad \dashrightarrow \pmb{\underline{\purple{\:2^{12} }} }\:\:\bigstar$

⠀⠀⠀⠀⠀$\therefore {\underline{ \sf \:Hence \:The \:value \:of\:\dfrac{8^x }{2^y}\:is\:\bf 2^{12}}}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\large {\boxed{\sf{\mid{\overline {\underline {\star More\:To\:know\::}}}\mid}}}$

$\qquad \qquad \boxed{\begin{array}{cc}\bf{\dag}\:\:\underline{\text{Law of Exponents :}}\\\\\bigstar\:\:\sf\dfrac{a^m}{a^n} = a^{m - n}\\\\\bigstar\:\:\sf{(a^m)^n = a^{mn}}\\\\\bigstar\:\:\sf(a^m)(a^n) = a^{m + n}\\\\\bigstar\:\:\sf\dfrac{1}{a^n} = a^{-n}\\\\\bigstar\:\:\sf\sqrt[\sf n]{\sf a} = (a)^{\dfrac{1}{n}}\end{array}}$

Answer 2

Answer:

Answers for x and y according to the first equation, 3x-y=12, are:

(3,-3), (4,0), (5,3) etc.

Plug them into the second equation, f(x,y)=(8^x)/(2^y):

f(3,-3) = (8^3)/(2^-3) = (8^3)/(1/8) = 8^4

f(4,0) = (8^4)/(2^0) = (8^4)/1 = 8^4

f(5,3) = (8^5)/(2^3) = (8^5)/8 = 8^4

The value is 8^4 but this can be simplified

8 = 2^3

(2^3)^4 = 2^12

= 4096