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. $$ \sqrt[3]{x^{2}} $$

Question

UpStudy AI · Accepted Answer

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!
Here are the simplified results and exponential forms: 1. **Expression:** $$ \frac{4^{x+2} \cdot 8^{x-1}}{2^{x-3} \cdot 16^{x}} = 16 $$ 2. **Expression:** $$ \frac{15^{n+2} \cdot 45^{1-n}}{3^{3-n}} = 375 $$ 3. **Expression:** $$ \frac{2^{x}+2^{x+1}}{2^{x}-2^{x+2}} = -1 $$ 4. **Expression:** $$ \frac{25^{x+1} \cdot 6}{10^{x-1} \cdot 15^{x}} = \frac{125}{2^{-2+x} \times 3^{x-1}} $$ 5. **Expression:** $$ \frac{6^{n+3} \cdot 2^{n-1}}{12^{n+2}} = \frac{3}{4} $$ 6. **Expression:** $$ \frac{2^{4 x+1} \cdot 9^{x} \cdot 6^{2 x-1}}{12^{3 x} \cdot 3^{x}} = \frac{1}{3} $$ **Conversions to Exponential Form:** 1. **Expression:** $$ \sqrt{x} = x^{\frac{1}{2}} $$ 2. **Expression:** $$ \sqrt[3]{x^{2}} = x^{\frac{2}{3}} $$

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