\( \left. \begin{array} { l l } { 3 } & { 3 ^ { x ^ { 2 } + 2 } = 9 \cdot \frac { 1 } { 27 ^ { x } } } \\ { 4 } & { 5 ^ { x ^ { 2 } - 1 } \cdot \frac { 1 } { 25 } = 25 ^ { x } } \\ { 5 } & { 2 ^ { 2 x } \cdot 4 ^ { x + 1 } = 1 } \\ { 6 } & { 12 ^ { x } \cdot 12 ^ { x } = 144 } \\ { 7 } & { 2 ^ { x ^ { 2 } \cdot 2 ^ { x ^ { 2 } - 2 } + 1 = 65 } } \\ { 8 } & { \frac { 4 ^ { x + 1 } \times ( \frac { 1 } { 4 } ) ^ { - x } } { 2 ^ { 2 x + 1 } } = \frac { 1 } { 32 } } \\ { 9 } & { 81 \cdot 9 ^ { x - 1 } = \frac { 1 } { 27 } } \\ { 10 } & { 10 ^ { 2 x ^ { 2 } } \cdot 10 ^ { x + 1 } = 10000 } \end{array} \right. \)

Solve the equation by following steps: - step0: Solve for \(x\): \(12^{x}\times 12^{x}=144\) - step1: Multiply the terms: \(12^{2x}=144\) - step2: Rewrite in exponential form: \(12^{2x}=12^{2}\) - step3: Set the exponents equal: \(2x=2\) - step4: Divide both sides: \(\frac{2x}{2}=\frac{2}{2}\) - step5: Divide the numbers: \(x=1\) Solve the equation \( 2^{2x} \cdot 4^{x+1}=1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{2x}\times 4^{x+1}=1\) - step1: Multiply the terms: \(2^{4x+2}=1\) - step2: Rewrite in exponential form: \(2^{4x+2}=2^{0}\) - step3: Set the exponents equal: \(4x+2=0\) - step4: Move the constant to the right side: \(4x=0-2\) - step5: Remove 0: \(4x=-2\) - step6: Divide both sides: \(\frac{4x}{4}=\frac{-2}{4}\) - step7: Divide the numbers: \(x=-\frac{1}{2}\) Solve the equation \( 81 \cdot 9^{x-1}=\frac{1}{27} \). Solve the equation by following steps: - step0: Solve for \(x\): \(81\times 9^{x-1}=\frac{1}{27}\) - step1: Multiply the terms: \(9^{x+1}=\frac{1}{27}\) - step2: Rewrite in exponential form: \(3^{2\left(x+1\right)}=3^{-3}\) - step3: Set the exponents equal: \(2\left(x+1\right)=-3\) - step4: Divide both sides: \(\frac{2\left(x+1\right)}{2}=\frac{-3}{2}\) - step5: Divide the numbers: \(x+1=-\frac{3}{2}\) - step6: Move the constant to the right side: \(x=-\frac{3}{2}-1\) - step7: Subtract the numbers: \(x=-\frac{5}{2}\) Solve the equation \( \frac{4^{x+1} \times (\frac{1}{4})^{-x}}{2^{2x+1}}=\frac{1}{32} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{4^{x+1}\left(\frac{1}{4}\right)^{-x}}{2^{2x+1}}=\frac{1}{32}\) - step1: Reduce the fraction: \(2^{2x+1}=\frac{1}{32}\) - step2: Rewrite in exponential form: \(2^{2x+1}=2^{-5}\) - step3: Set the exponents equal: \(2x+1=-5\) - step4: Move the constant to the right side: \(2x=-5-1\) - step5: Subtract the numbers: \(2x=-6\) - step6: Divide both sides: \(\frac{2x}{2}=\frac{-6}{2}\) - step7: Divide the numbers: \(x=-3\) Solve the equation \( 3^{x^{2}+2}=9 \cdot \frac{1}{27^{x}} \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x^{2}+2}=9\times \frac{1}{27^{x}}\) - step1: Simplify: \(3^{x^{2}+2}=3^{-3x+2}\) - step2: Set the exponents equal: \(x^{2}+2=-3x+2\) - step3: Cancel equal terms: \(x^{2}=-3x\) - step4: Move the expression to the left side: \(x^{2}-\left(-3x\right)=0\) - step5: Add or subtract both sides: \(x^{2}+3x=0\) - step6: Factor the expression: \(x\left(x+3\right)=0\) - step7: Separate into possible cases: \(\begin{align}&x=0\\&x+3=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=0\\&x=-3\end{align}\) - step9: Rewrite: \(x_{1}=-3,x_{2}=0\) Solve the equation \( 5^{x^{2}-1} \cdot \frac{1}{25}=25^{x} \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{x^{2}-1}\times \frac{1}{25}=25^{x}\) - step1: Multiply the terms: \(5^{x^{2}-3}=25^{x}\) - step2: Rewrite the expression: \(5^{x^{2}-3}=5^{2x}\) - step3: Set the exponents equal: \(x^{2}-3=2x\) - step4: Move the expression to the left side: \(x^{2}-3-2x=0\) - step5: Factor the expression: \(\left(x-3\right)\left(x+1\right)=0\) - step6: Separate into possible cases: \(\begin{align}&x-3=0\\&x+1=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=3\\&x=-1\end{align}\) - step8: Rewrite: \(x_{1}=-1,x_{2}=3\) Solve the equation \( 10^{2x^{2}} \cdot 10^{x+1}=10000 \). Solve the equation by following steps: - step0: Solve for \(x\): \(10^{2x^{2}}\times 10^{x+1}=10000\) - step1: Multiply: \(10^{2x^{2}+x+1}=10000\) - step2: Rewrite in exponential form: \(10^{2x^{2}+x+1}=10^{4}\) - step3: Set the exponents equal: \(2x^{2}+x+1=4\) - step4: Move the expression to the left side: \(2x^{2}+x+1-4=0\) - step5: Subtract the numbers: \(2x^{2}+x-3=0\) - step6: Factor the expression: \(\left(x-1\right)\left(2x+3\right)=0\) - step7: Separate into possible cases: \(\begin{align}&x-1=0\\&2x+3=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=1\\&x=-\frac{3}{2}\end{align}\) - step9: Rewrite: \(x_{1}=-\frac{3}{2},x_{2}=1\) Here are the solutions for each of the equations provided: 1. **Equation:** \( 3^{x^{2}+2} = 9 \cdot \frac{1}{27^{x}} \) **Solution:** \[ x_{1} = -3, \quad x_{2} = 0 \] 2. **Equation:** \( 5^{x^{2}-1} \cdot \frac{1}{25} = 25^{x} \) **Solution:** \[ x_{1} = -1, \quad x_{2} = 3 \] 3. **Equation:** \( 2^{2x} \cdot 4^{x+1} = 1 \) **Solution:** \[ x = -\frac{1}{2} \quad \text{(or } x = -0.5\text{)} \] 4. **Equation:** \( 12^{x} \cdot 12^{x} = 144 \) **Solution:** \[ x = 1 \] 5. **Equation:** \( 2^{x^{2} \cdot 2^{x^{2}-2}} = 65 \) **Solution:** \[ \text{Error: Unsupported calculation} \] 6. **Equation:** \( \frac{4^{x+1} \times \left(\frac{1}{4}\right)^{-x}}{2^{2x+1}} = \frac{1}{32} \) **Solution:** \[ x = -3 \] 7. **Equation:** \( 81 \cdot 9^{x-1} = \frac{1}{27} \) **Solution:** \[ x = -\frac{5}{2} \quad \text{(or } x = -2.5\text{)} \] 8. **Equation:** \( 10^{2x^{2}} \cdot 10^{x+1} = 10000 \) **Solution:** \[ x_{1} = -\frac{3}{2}, \quad x_{2} = 1 \quad \text{(or } x_{1} = -1.5, x_{2} = 1\text{)} \] If you need further assistance or explanations for any specific equation, feel free to ask!