Solve for $$ x $$ i $$ 3^{x}-3^{x-2}=24 $$ if $$ 3^{x}+3^{x+2}=10 $$ iii $$ 2^{x}+2^{x}+2^{x}=96 $$ iv $$ 12^{x}+12^{x}=288 $$ $$ 2^{x}+4^{x+1}=80 $$

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Solve the equation $$ 3^{x}-3^{x-2}=24 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$3^{x}-3^{x-2}=24$$
- step1: Subtract the terms:
  $$8	imes 3^{x-2}=24$$
- step2: Divide both sides:
  $$\frac{8	imes 3^{x-2}}{8}=\frac{24}{8}$$
- step3: Divide the numbers:
  $$3^{x-2}=3$$
- step4: Set the exponents equal:
  $$x-2=1$$
- step5: Move the constant to the right side:
  $$x=1+2$$
- step6: Add the numbers:
  $$x=3$$
Solve the equation $$ 3^{x}+3^{x+2}=10 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$3^{x}+3^{x+2}=10$$
- step1: Add the terms:
  $$10	imes 3^{x}=10$$
- step2: Divide both sides:
  $$\frac{10	imes 3^{x}}{10}=\frac{10}{10}$$
- step3: Divide the numbers:
  $$3^{x}=1$$
- step4: Rewrite in exponential form:
  $$3^{x}=3^{0}$$
- step5: Set the exponents equal:
  $$x=0$$
Solve the equation $$ 2^{x}+2^{x}+2^{x}=96 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2^{x}+2^{x}+2^{x}=96$$
- step1: Simplify:
  $$3	imes 2^{x}=96$$
- step2: Divide both sides:
  $$\frac{3	imes 2^{x}}{3}=\frac{96}{3}$$
- step3: Divide the numbers:
  $$2^{x}=32$$
- step4: Rewrite in exponential form:
  $$2^{x}=2^{5}$$
- step5: Set the exponents equal:
  $$x=5$$
Solve the equation $$ 12^{x}+12^{x}=288 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$12^{x}+12^{x}=288$$
- step1: Add the terms:
  $$2	imes 12^{x}=288$$
- step2: Divide both sides:
  $$\frac{2	imes 12^{x}}{2}=\frac{288}{2}$$
- step3: Divide the numbers:
  $$12^{x}=144$$
- step4: Rewrite in exponential form:
  $$12^{x}=12^{2}$$
- step5: Set the exponents equal:
  $$x=2$$
Solve the equation $$ 2^{x}+4^{x+1}=80 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2^{x}+4^{x+1}=80$$
- step1: Move the expression to the left side:
  $$2^{x}+4^{x+1}-80=0$$
- step2: Rewrite the expression:
  $$2^{x}+4	imes 4^{x}-80=0$$
- step3: Rewrite the expression:
  $$2^{x}+4	imes 2^{2x}-80=0$$
- step4: Use substitution:
  $$t+4t^{2}-80=0$$
- step5: Rewrite in standard form:
  $$4t^{2}+t-80=0$$
- step6: Solve using the quadratic formula:
  $$t=\frac{-1\pm \sqrt{1^{2}-4	imes 4\left(-80ight)}}{2	imes 4}$$
- step7: Simplify the expression:
  $$t=\frac{-1\pm \sqrt{1^{2}-4	imes 4\left(-80ight)}}{8}$$
- step8: Simplify the expression:
  $$t=\frac{-1\pm \sqrt{1281}}{8}$$
- step9: Separate into possible cases:
  $$\begin{align}&t=\frac{-1+\sqrt{1281}}{8}\&t=\frac{-1-\sqrt{1281}}{8}\end{align}$$
- step10: Rewrite the fraction:
  $$\begin{align}&t=\frac{-1+\sqrt{1281}}{8}\&t=-\frac{1+\sqrt{1281}}{8}\end{align}$$
- step11: Substitute back:
  $$\begin{align}&2^{x}=\frac{-1+\sqrt{1281}}{8}\&2^{x}=-\frac{1+\sqrt{1281}}{8}\end{align}$$
- step12: Solve the equation for $$x:$$
  $$\begin{align}&x=\log_{2}{\left(-1+\sqrt{1281}ight)}-3\&x 
otin \mathbb{R}\end{align}$$
- step13: Find the union:
  $$x=\log_{2}{\left(-1+\sqrt{1281}ight)}-3$$
The solutions to the given equations are:
1. $$ x = 3 $$
2. $$ x = 0 $$
3. $$ x = 5 $$
4. $$ x = 2 $$
5. $$ x = \log_{2}(-1+\sqrt{1281})-3 $$
The solutions are: 1. $$ x = 3 $$ 2. $$ x = 0 $$ 3. $$ x = 5 $$ 4. $$ x = 2 $$ 5. $$ x = \log_{2}(-1+\sqrt{1281})-3 $$

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