1.1 Simplify \( 1.1 .1 \frac{4^{x+2} \cdot 8^{x-1}}{2^{x-3} \cdot 16^{x}} \) 1.1.2. \( \frac{15^{n+2} \cdot 45^{1-n}}{3^{3-n}} \) \( 11.3 \frac{2^{x}+2^{x+1}}{2^{x}-2^{x+2}} \) 1.1.4 \( \frac{25^{x+1} \cdot 6}{10^{x-1} \cdot 15^{x}} \) \( 1.1 .5 \quad \frac{6^{n+3} \cdot 2^{n-1}}{12^{n+2}} \) 1.1.6 \( \frac{2^{4 x+1} \cdot 9^{x} \cdot 6^{2 x-1}}{12^{3 x} \cdot 3^{x}} \) 1.2 Convert the following into exponential form 1.2.1 a. \( \sqrt{x} \) b. \( \sqrt[3]{x^{2}} \)

Simplify the expression by following steps: - step0: Solution: \(\sqrt[3]{x^{2}}\) Calculate or simplify the expression \( \frac{15^{n+2} \cdot 45^{1-n}}{3^{3-n}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{15^{n+2}\times 45^{1-n}}{3^{3-n}}\) - step1: Factor the expression: \(\frac{3^{n+2}\times 5^{n+2}\times 45^{1-n}}{3^{-n+3}}\) - step2: Reduce the fraction: \(3^{2n-1}\times 5^{n+2}\times 45^{1-n}\) - step3: Rewrite the expression: \(3\times 5^{1-n}\times 5^{n+2}\) - step4: Multiply the terms: \(3\times 5^{3}\) - step5: Multiply the terms: \(375\) Calculate or simplify the expression \( \frac{25^{x+1} \cdot 6}{10^{x-1} \cdot 15^{x}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{25^{x+1}\times 6}{10^{x-1}\times 15^{x}}\) - step1: Factor the expression: \(\frac{5^{2x+2}\times 6}{5^{x-1}\times 2^{x-1}\times 15^{x}}\) - step2: Reduce the fraction: \(\frac{5^{x+3}\times 6}{2^{x-1}\times 15^{x}}\) - step3: Factor the expression: \(\frac{5^{x+3}\times 6}{2^{x-1}\times 5^{x}\times 3^{x}}\) - step4: Reduce the fraction: \(\frac{5^{3}\times 6}{2^{x-1}\times 3^{x}}\) - step5: Factor the expression: \(\frac{5^{3}\times 6}{2^{-1+x}\times 3^{x}}\) - step6: Calculate: \(\frac{750}{2^{-1+x}\times 3^{x}}\) - step7: Calculate: \(\frac{125\times 6}{2^{-2+x}\times 3^{x-1}\times 6}\) - step8: Reduce the fraction: \(\frac{125}{2^{-2+x}\times 3^{x-1}}\) Calculate or simplify the expression \( \frac{2^{x}+2^{x+1}}{2^{x}-2^{x+2}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2^{x}+2^{x+1}}{2^{x}-2^{x+2}}\) - step1: Add the terms: \(\frac{3\times 2^{x}}{2^{x}-2^{x+2}}\) - step2: Subtract the terms: \(\frac{3\times 2^{x}}{-3\times 2^{x}}\) - step3: Rewrite the fraction: \(-\frac{3\times 2^{x}}{3\times 2^{x}}\) - step4: Reduce the fraction: \(-1\) Calculate or simplify the expression \( \frac{4^{x+2} \cdot 8^{x-1}}{2^{x-3} \cdot 16^{x}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{4^{x+2}\times 8^{x-1}}{2^{x-3}\times 16^{x}}\) - step1: Multiply: \(\frac{2^{5x+1}}{2^{x-3}\times 16^{x}}\) - step2: Multiply: \(\frac{2^{5x+1}}{2^{5x-3}}\) - step3: Divide the numbers: \(2^{5x+1-\left(5x-3\right)}\) - step4: Subtract the terms: \(2^{4}\) - step5: Evaluate the power: \(16\) Calculate or simplify the expression \( \frac{6^{n+3} \cdot 2^{n-1}}{12^{n+2}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{6^{n+3}\times 2^{n-1}}{12^{n+2}}\) - step1: Factor the expression: \(\frac{6^{n+3}\times 2^{n-1}}{6^{n+2}\times 2^{n+2}}\) - step2: Reduce the fraction: \(\frac{6}{2^{3}}\) - step3: Factor the expression: \(\frac{2\times 3}{2^{3}}\) - step4: Reduce the fraction: \(\frac{3}{2^{2}}\) - step5: Evaluate the power: \(\frac{3}{4}\) Calculate or simplify the expression \( \sqrt{x} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{x}\) Calculate or simplify the expression \( \frac{2^{4 x+1} \cdot 9^{x} \cdot 6^{2 x-1}}{12^{3 x} \cdot 3^{x}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2^{4x+1}\times 9^{x}\times 6^{2x-1}}{12^{3x}\times 3^{x}}\) - step1: Factor the expression: \(\frac{2^{4x+1}\times 9^{x}\times 6^{2x-1}}{2^{4x}\times 108^{x}\times 3^{x}}\) - step2: Reduce the fraction: \(\frac{2\times 9^{x}\times 6^{2x-1}}{108^{x}\times 3^{x}}\) - step3: Factor the expression: \(\frac{2\times 9^{x}\times 6^{2x-1}}{2^{2x}\times 27^{x}\times 3^{x}}\) - step4: Divide the numbers: \(\frac{2^{1-2x}\times 9^{x}\times 6^{2x-1}}{27^{x}\times 3^{x}}\) - step5: Factor the expression: \(\frac{2^{1-2x}\times 3^{2x}\times 6^{2x-1}}{3^{3x}\times 3^{x}}\) - step6: Reduce the fraction: \(\frac{2^{1-2x}\times 6^{2x-1}}{3^{x}\times 3^{x}}\) - step7: Factor the expression: \(\frac{2^{1-2x}\times 3^{2x-1}\times 2^{2x-1}}{3^{x}\times 3^{x}}\) - step8: Reduce the fraction: \(\frac{2^{1-2x}\times 2^{2x-1}}{3}\) - step9: Calculate: \(\frac{1}{3}\) Here are the simplified results for each expression and the conversions to exponential form: ### Simplifications 1. **Expression:** \[ \frac{4^{x+2} \cdot 8^{x-1}}{2^{x-3} \cdot 16^{x}} \] **Simplified Result:** \[ 16 \] 2. **Expression:** \[ \frac{15^{n+2} \cdot 45^{1-n}}{3^{3-n}} \] **Simplified Result:** \[ 375 \] 3. **Expression:** \[ \frac{2^{x}+2^{x+1}}{2^{x}-2^{x+2}} \] **Simplified Result:** \[ -1 \] 4. **Expression:** \[ \frac{25^{x+1} \cdot 6}{10^{x-1} \cdot 15^{x}} \] **Simplified Result:** \[ \frac{125}{2^{-2+x} \times 3^{x-1}} \] 5. **Expression:** \[ \frac{6^{n+3} \cdot 2^{n-1}}{12^{n+2}} \] **Simplified Result:** \[ \frac{3}{4} \] 6. **Expression:** \[ \frac{2^{4 x+1} \cdot 9^{x} \cdot 6^{2 x-1}}{12^{3 x} \cdot 3^{x}} \] **Simplified Result:** \[ \frac{1}{3} \] ### Conversions to Exponential Form 1. **Expression:** \[ \sqrt{x} \] **Exponential Form:** \[ x^{\frac{1}{2}} \] 2. **Expression:** \[ \sqrt[3]{x^{2}} \] **Exponential Form:** \[ x^{\frac{2}{3}} \] If you have any further questions or need additional assistance, feel free to ask!