$$ 5 \quad(5)^{x}-(5)^{-x-2}=24 {} 6 \quad 2^{\frac{x}{3}}+2^{\frac{x}{3}+1}=12 {} 7\left(5^{x}+10\right)\left(2^{x}-0,0625\right)=0 {} 8 \quad 2^{2 x}-3.2^{x}=-2 {} 9 \quad 9^{x}+6 \cdot 3^{x}=27 {} 103^{2+x}+3^{2-x}=82 {} 115^{x+1}+5^{1+x}=26 {} 122^{x+1}+2^{3-x}=17 {} 133^{x+2}-3^{x-2}=80 {} 142^{x}=\frac{27^{\frac{1}{2}}}{3^{\frac{21}{2}}+3^{1 \frac{1}{2}}} {} 15.6 x-5 x^{\frac{1}{2}}=6 {} 167^{x}-50=-49.7^{-x} {} 174.7^{4 x}=49.2^{4 x} {} 18 \quad 3^{3 x} \cdot 16=2^{3 x} \cdot 81 $$

Question

UpStudy AI · Accepted Answer

Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\left(5^{x}+10\right)\left(2^{x}-0.0625\right)=0$$
- step1: Separate into possible cases:
  $$\begin{align}&5^{x}+10=0\&2^{x}-0.0625=0\end{align}$$
- step2: Solve the equation:
  $$\begin{align}&x \notin \mathbb{R}\&x=-4\end{align}$$
- step3: Find the union:
  $$x=-4$$
Solve the equation $$ 6 x-5 x^{\frac{1}{2}}=6 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$6x-5x^{\frac{1}{2}}=6$$
- step1: Find the domain:
  $$6x-5x^{\frac{1}{2}}=6,x\geq 0$$
- step2: Move the expression to the left side:
  $$6x-5x^{\frac{1}{2}}-6=0$$
- step3: Solve using substitution:
  $$6t^{2}-5t-6=0$$
- step4: Factor the expression:
  $$\left(2t-3\right)\left(3t+2\right)=0$$
- step5: Separate into possible cases:
  $$\begin{align}&2t-3=0\&3t+2=0\end{align}$$
- step6: Solve the equation:
  $$\begin{align}&t=\frac{3}{2}\&t=-\frac{2}{3}\end{align}$$
- step7: Substitute back:
  $$\begin{align}&x^{\frac{1}{2}}=\frac{3}{2}\&x^{\frac{1}{2}}=-\frac{2}{3}\end{align}$$
- step8: Solve the equation for $$x:$$
  $$\begin{align}&x=\frac{9}{4}\&x \notin \mathbb{R}\end{align}$$
- step9: Find the union:
  $$x=\frac{9}{4}$$
- step10: Check if the solution is in the defined range:
  $$x=\frac{9}{4},x\geq 0$$
- step11: Find the intersection:
  $$x=\frac{9}{4}$$
Solve the equation $$ 5^{x+1}+5^{1+x}=26 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$5^{x+1}+5^{1+x}=26$$
- step1: Evaluate:
  $$2\times 5^{1+x}=26$$
- step2: Divide both sides:
  $$\frac{2\times 5^{1+x}}{2}=\frac{26}{2}$$
- step3: Divide the numbers:
  $$5^{1+x}=13$$
- step4: Take the logarithm of both sides:
  $$\log_{5}{\left(5^{1+x}\right)}=\log_{5}{\left(13\right)}$$
- step5: Evaluate the logarithm:
  $$1+x=\log_{5}{\left(13\right)}$$
- step6: Move the constant to the right side:
  $$x=\log_{5}{\left(13\right)}-1$$
Solve the equation $$ 9^{x}+6 \cdot 3^{x}=27 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$9^{x}+6\times 3^{x}=27$$
- step1: Move the expression to the left side:
  $$9^{x}+6\times 3^{x}-27=0$$
- step2: Factor the expression:
  $$\left(3^{x}-3\right)\left(3^{x}+9\right)=0$$
- step3: Separate into possible cases:
  $$\begin{align}&3^{x}-3=0\&3^{x}+9=0\end{align}$$
- step4: Solve the equation:
  $$\begin{align}&x=1\&x \notin \mathbb{R}\end{align}$$
- step5: Find the union:
  $$x=1$$
Solve the equation $$ 3^{3 x} \cdot 16=2^{3 x} \cdot 81 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$3^{3x}\times 16=2^{3x}\times 81$$
- step1: Multiply the terms:
  $$16\times 3^{3x}=81\times 2^{3x}$$
- step2: Multiply both sides:
  $$\frac{1}{16}\times 16\times 3^{3x}=\frac{1}{16}\times 81\times 2^{3x}$$
- step3: Calculate:
  $$3^{3x}=\frac{81}{16}\times 2^{3x}$$
- step4: Divide both sides:
  $$\frac{3^{3x}}{2^{3x}}=\frac{81}{16}$$
- step5: Evaluate:
  $$\left(\frac{3}{2}\right)^{3x}=\frac{81}{16}$$
- step6: Rewrite in exponential form:
  $$\left(\frac{3}{2}\right)^{3x}=\left(\frac{3}{2}\right)^{4}$$
- step7: Set the exponents equal:
  $$3x=4$$
- step8: Divide both sides:
  $$\frac{3x}{3}=\frac{4}{3}$$
- step9: Divide the numbers:
  $$x=\frac{4}{3}$$
Solve the equation $$ 2^{x+1}+2^{3-x}=17 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2^{x+1}+2^{3-x}=17$$
- step1: Move the expression to the left side:
  $$2^{x+1}+2^{3-x}-17=0$$
- step2: Factor the expression:
  $$\left(2^{x}-8\right)\left(2^{x+1}-1\right)\left(2^{x}\right)^{-1}=0$$
- step3: Rewrite the expression:
  $$\frac{2^{2x+1}-136\times 2^{x-3}+8}{2^{x}}=0$$
- step4: Cross multiply:
  $$2^{2x+1}-136\times 2^{x-3}+8=2^{x}\times 0$$
- step5: Simplify the equation:
  $$2^{2x+1}-136\times 2^{x-3}+8=0$$
- step6: Factor the expression:
  $$\left(2^{x}-8\right)\left(2^{x+1}-1\right)=0$$
- step7: Separate into possible cases:
  $$\begin{align}&2^{x}-8=0\&2^{x+1}-1=0\end{align}$$
- step8: Solve the equation:
  $$\begin{align}&x=3\&x=-1\end{align}$$
- step9: Rewrite:
  $$x_{1}=-1,x_{2}=3$$
Solve the equation $$ 2^{x}=\frac{27^{\frac{1}{2}}}{3^{\frac{21}{2}}+3^{1 \frac{1}{2}}} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2^{x}=\frac{27^{\frac{1}{2}}}{3^{\frac{21}{2}}+3^{1\frac{1}{2}}}$$
- step1: Simplify:
  $$2^{x}=\frac{1}{19684}$$
- step2: Take the logarithm of both sides:
  $$\log_{2}{\left(2^{x}\right)}=\log_{2}{\left(\frac{1}{19684}\right)}$$
- step3: Evaluate the logarithm:
  $$x=\log_{2}{\left(\frac{1}{19684}\right)}$$
- step4: Simplify:
  $$x=-\log_{2}{\left(19684\right)}$$
- step5: Simplify:
  $$x=-2-\log_{2}{\left(4921\right)}$$
Solve the equation $$ 2^{\frac{x}{3}}+2^{\frac{x}{3}+1}=12 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2^{\frac{x}{3}}+2^{\frac{x}{3}+1}=12$$
- step1: Add the terms:
  $$3\times 2^{\frac{x}{3}}=12$$
- step2: Divide both sides:
  $$\frac{3\times 2^{\frac{x}{3}}}{3}=\frac{12}{3}$$
- step3: Divide the numbers:
  $$2^{\frac{x}{3}}=4$$
- step4: Rewrite in exponential form:
  $$2^{\frac{x}{3}}=2^{2}$$
- step5: Set the exponents equal:
  $$\frac{x}{3}=2$$
- step6: Cross multiply:
  $$x=3\times 2$$
- step7: Simplify the equation:
  $$x=6$$
Solve the equation $$ 3^{x+2}-3^{x-2}=80 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$3^{x+2}-3^{x-2}=80$$
- step1: Subtract the terms:
  $$80\times 3^{x-2}=80$$
- step2: Divide both sides:
  $$\frac{80\times 3^{x-2}}{80}=\frac{80}{80}$$
- step3: Divide the numbers:
  $$3^{x-2}=1$$
- step4: Rewrite in exponential form:
  $$3^{x-2}=3^{0}$$
- step5: Set the exponents equal:
  $$x-2=0$$
- step6: Move the constant to the right side:
  $$x=0+2$$
- step7: Remove 0:
  $$x=2$$
Solve the equation $$ 7^{4 x}=49^{4 x} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$7^{4x}=49^{4x}$$
- step1: Rewrite the expression:
  $$7^{4x}=7^{8x}$$
- step2: Set the exponents equal:
  $$4x=8x$$
- step3: Add or subtract both sides:
  $$4x-8x=0$$
- step4: Subtract the terms:
  $$-4x=0$$
- step5: Change the signs:
  $$4x=0$$
- step6: Rewrite the expression:
  $$x=0$$
Solve the equation $$ (5)^{x}-(5)^{-x-2}=24 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$5^{x}-5^{-x-2}=24$$
- step1: Move the expression to the left side:
  $$5^{x}-5^{-x-2}-24=0$$
- step2: Factor the expression:
  $$\frac{1}{25}\left(5^{2x+2}-1-600\times 5^{x}\right)\left(5^{x}\right)^{-1}=0$$
- step3: Rewrite the expression:
  $$\frac{5^{2x+2}-1-600\times 5^{x}}{5^{x+2}}=0$$
- step4: Cross multiply:
  $$5^{2x+2}-1-600\times 5^{x}=5^{x+2}\times 0$$
- step5: Simplify the equation:
  $$5^{2x+2}-1-600\times 5^{x}=0$$
- step6: Rewrite the expression:
  $$25\times 5^{2x}-1-600\times 5^{x}=0$$
- step7: Use substitution:
  $$25t^{2}-1-600t=0$$
- step8: Rewrite in standard form:
  $$25t^{2}-600t-1=0$$
- step9: Solve using the quadratic formula:
  $$t=\frac{600\pm \sqrt{\left(-600\right)^{2}-4\times 25\left(-1\right)}}{2\times 25}$$
- step10: Simplify the expression:
  $$t=\frac{600\pm \sqrt{\left(-600\right)^{2}-4\times 25\left(-1\right)}}{50}$$
- step11: Simplify the expression:
  $$t=\frac{600\pm \sqrt{600^{2}+100}}{50}$$
- step12: Simplify the expression:
  $$t=\frac{600\pm 10\sqrt{3601}}{50}$$
- step13: Separate into possible cases:
  $$\begin{align}&t=\frac{600+10\sqrt{3601}}{50}\&t=\frac{600-10\sqrt{3601}}{50}\end{align}$$
- step14: Simplify the expression:
  $$\begin{align}&t=\frac{60+\sqrt{3601}}{5}\&t=\frac{600-10\sqrt{3601}}{50}\end{align}$$
- step15: Simplify the expression:
  $$\begin{align}&t=\frac{60+\sqrt{3601}}{5}\&t=\frac{60-\sqrt{3601}}{5}\end{align}$$
- step16: Substitute back:
  $$\begin{align}&5^{x}=\frac{60+\sqrt{3601}}{5}\&5^{x}=\frac{60-\sqrt{3601}}{5}\end{align}$$
- step17: Solve the equation for $$x:$$
  $$\begin{align}&x=\log_{5}{\left(60+\sqrt{3601}\right)}-1\&x \notin \mathbb{R}\end{align}$$
- step18: Find the union:
  $$x=\log_{5}{\left(60+\sqrt{3601}\right)}-1$$
Solve the equation $$ 2^{2 x}-3 \cdot 2^{x}=-2 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2^{2x}-3\times 2^{x}=-2$$
- step1: Move the expression to the left side:
  $$2^{2x}-3\times 2^{x}-\left(-2\right)=0$$
- step2: Remove the parentheses:
  $$2^{2x}-3\times 2^{x}+2=0$$
- step3: Factor the expression:
  $$\left(2^{x}-2\right)\left(2^{x}-1\right)=0$$
- step4: Separate into possible cases:
  $$\begin{align}&2^{x}-2=0\&2^{x}-1=0\end{align}$$
- step5: Solve the equation:
  $$\begin{align}&x=1\&x=0\end{align}$$
- step6: Rewrite:
  $$x_{1}=0,x_{2}=1$$
Solve the equation $$ 3^{2+x}+3^{2-x}=82 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$3^{2+x}+3^{2-x}=82$$
- step1: Move the expression to the left side:
  $$3^{2+x}+3^{2-x}-82=0$$
- step2: Factor the expression:
  $$\left(3^{x}-9\right)\left(3^{x+2}-1\right)\left(3^{x}\right)^{-1}=0$$
- step3: Rewrite the expression:
  $$\frac{3^{2x+2}-738\times 3^{x-2}+9}{3^{x}}=0$$
- step4: Cross multiply:
  $$3^{2x+2}-738\times 3^{x-2}+9=3^{x}\times 0$$
- step5: Simplify the equation:
  $$3^{2x+2}-738\times 3^{x-2}+9=0$$
- step6: Factor the expression:
  $$\left(3^{x}-9\right)\left(3^{x+2}-1\right)=0$$
- step7: Separate into possible cases:
  $$\begin{align}&3^{x}-9=0\&3^{x+2}-1=0\end{align}$$
- step8: Solve the equation:
  $$\begin{align}&x=2\&x=-2\end{align}$$
- step9: Rewrite:
  $$x_{1}=-2,x_{2}=2$$
Here are the solutions to the equations provided:

1. **Equation:** $$ (5)^{x} - (5)^{-x-2} = 24 $$  
   **Solution:** 
   $$
   x = \log_{5}{(60+\sqrt{3601})} - 1
   $$

2. **Equation:** $$ 2^{\frac{x}{3}} + 2^{\frac{x}{3}+1} = 12 $$  
   **Solution:** 
   $$
   x = 6
   $$

3. **Equation:** $$ (5^{x} + 10)(2^{x} - 0.0625) = 0 $$  
   **Solution:** 
   $$
   x = -4
   $$

4. **Equation:** $$ 2^{2x} - 3 \cdot 2^{x} = -2 $$  
   **Solution:** 
   $$
   x_{1} = 0, \quad x_{2} = 1
   $$

5. **Equation:** $$ 9^{x} + 6 \cdot 3^{x} = 27 $$  
   **Solution:** 
   $$
   x = 1
   $$

6. **Equation:** $$ 3^{2+x} + 3^{2-x} = 82 $$  
   **Solution:** 
   $$
   x_{1} = -2, \quad x_{2} = 2
   $$

7. **Equation:** $$ 5^{x+1} + 5^{1+x} = 26 $$  
   **Solution:** 
   $$
   x = \log_{5}{(13)} - 1
   $$

8. **Equation:** $$ 2^{x+1} + 2^{3-x} = 17 $$  
   **Solution:** 
   $$
   x_{1} = -1, \quad x_{2} = 3
   $$

9. **Equation:** $$ 3^{x+2} - 3^{x-2} = 80 $$  
   **Solution:** 
   $$
   x = 2
   $$

10. **Equation:** $$ 2^{x} = \frac{27^{\frac{1}{2}}}{3^{\frac{21}{2}} + 3^{1 \frac{1}{2}}} $$  
    **Solution:** 
    $$
    x = -2 - \log_{2}{(4921)}
    $$

11. **Equation:** $$ 6x - 5x^{\frac{1}{2}} = 6 $$  
    **Solution:** 
    $$
    x = \frac{9}{4} \quad \text{or} \quad x = 2.25
    $$

12. **Equation:** $$ 7^{x} - 50 = -49.7^{-x} $$  
    **Solution:** 
    $$
    x = 0
    $$

13. **Equation:** $$ 7^{4x} = 49^{4x} $$  
    **Solution:** 
    $$
    x = \frac{4}{3} \quad \text{or} \quad x = 1.\dot{3}
    $$

14. **Equation:** $$ 3^{3x} \cdot 16 = 2^{3x} \cdot 81 $$  
    **Solution:** 
    $$
    \text{Unsupported calculation}
    $$

If you need further assistance or clarification on any of these solutions, feel free to ask!
Here are the solutions to the equations: 1. $$ x = \log_{5}{(60+\sqrt{3601})} - 1 $$ 2. $$ x = 6 $$ 3. $$ x = -4 $$ 4. $$ x_{1} = 0, \quad x_{2} = 1 $$ 5. $$ x = 1 $$ 6. $$ x_{1} = -2, \quad x_{2} = 2 $$ 7. $$ x = \log_{5}{(13)} - 1 $$ 8. $$ x_{1} = -1, \quad x_{2} = 3 $$ 9. $$ x = 2 $$ 10. $$ x = -2 - \log_{2}{(4921)} $$ 11. $$ x = \frac{9}{4} $$ or $$ x = 2.25 $$ 12. $$ x = 0 $$ 13. $$ x = \frac{4}{3} $$ or $$ x = 1.\dot{3} $$ 14. $$ \text{Unsupported calculation} $$ If you need more help with any of these, let me know!

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