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

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!