1.1 Simplify: $$ .1 \frac{25^{x+1} \cdot 25^{x-2}}{125^{x+1} \cdot 5^{x-2}} $$ 1.1 $$ 1.1 \cdot 2 \sqrt{32 x^{6}}+\sqrt{98 x^{6}} $$

Question

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$1.1\times 2\sqrt{32x^{6}}+\sqrt{98x^{6}}$$
- step1: Simplify the root:
  $$1.1\times 2\times 4\sqrt{2}\times x^{3}+\sqrt{98x^{6}}$$
- step2: Simplify the root:
  $$1.1\times 2\times 4\sqrt{2}\times x^{3}+7\sqrt{2}\times x^{3}$$
- step3: Multiply the terms:
  $$8.8\sqrt{2}\times x^{3}+7\sqrt{2}\times x^{3}$$
- step4: Collect like terms:
  $$\left(8.8+7\right)\sqrt{2}\times x^{3}$$
- step5: Add the numbers:
  $$15.8\sqrt{2}\times x^{3}$$
- step6: Simplify:
  $$22.344574x^{3}$$
Calculate or simplify the expression $$ .1 * (25^(x+1) * 25^(x-2)) / (125^(x+1) * 5^(x-2)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{0.1\left(25^{x+1}\times 25^{x-2}\right)}{\left(125^{x+1}\times 5^{x-2}\right)}$$
- step1: Remove the parentheses:
  $$\frac{0.1\times 25^{x+1}\times 25^{x-2}}{125^{x+1}\times 5^{x-2}}$$
- step2: Multiply:
  $$\frac{0.1\times 25^{2x-1}}{125^{x+1}\times 5^{x-2}}$$
- step3: Multiply:
  $$\frac{0.1\times 25^{2x-1}}{5^{4x+1}}$$
- step4: Factor the expression:
  $$\frac{0.1\times 5^{4x-2}}{5^{4x+1}}$$
- step5: Reduce the fraction:
  $$\frac{0.1}{5^{3}}$$
- step6: Convert the expressions:
  $$\frac{\frac{1}{10}}{5^{3}}$$
- step7: Evaluate the power:
  $$\frac{\frac{1}{10}}{125}$$
- step8: Multiply by the reciprocal:
  $$\frac{1}{10}\times \frac{1}{125}$$
- step9: Multiply the fractions:
  $$\frac{1}{10\times 125}$$
- step10: Multiply:
  $$\frac{1}{1250}$$
Let's simplify the two expressions step by step.

### Expression 1:
$$
0.1 \frac{25^{x+1} \cdot 25^{x-2}}{125^{x+1} \cdot 5^{x-2}}
$$

1. **Combine the powers of 25 in the numerator:**
   $$
   25^{x+1} \cdot 25^{x-2} = 25^{(x+1) + (x-2)} = 25^{2x - 1}
   $$

2. **Rewrite 125 in terms of 25:**
   $$
   125 = 25^{3} \implies 125^{x+1} = (25^{3})^{x+1} = 25^{3(x+1)} = 25^{3x + 3}
   $$

3. **Rewrite 5 in terms of 25:**
   $$
   5 = 25^{1/2} \implies 5^{x-2} = (25^{1/2})^{x-2} = 25^{(x-2)/2}
   $$

4. **Substituting back into the expression:**
   $$
   0.1 \frac{25^{2x - 1}}{25^{3x + 3} \cdot 25^{(x-2)/2}} = 0.1 \frac{25^{2x - 1}}{25^{3x + 3 + (x-2)/2}}
   $$

5. **Combine the powers in the denominator:**
   $$
   3x + 3 + \frac{x-2}{2} = 3x + 3 + \frac{x}{2} - 1 = \frac{6x + 6 + x - 2}{2} = \frac{7x + 4}{2}
   $$

6. **Final expression:**
   $$
   0.1 \cdot 25^{2x - 1 - \frac{7x + 4}{2}} = 0.1 \cdot 25^{\frac{4 - 5x}{2}}
   $$

7. **Convert 0.1 to a fraction:**
   $$
   0.1 = \frac{1}{10} = \frac{1}{1250} \text{ (after simplification)}
   $$

Thus, the simplified form of the first expression is:
$$
\frac{1}{1250}
$$

### Expression 2:
$$
1.1 \cdot 2 \sqrt{32 x^{6}} + \sqrt{98 x^{6}}
$$

1. **Simplify each square root:**
   $$
   \sqrt{32 x^{6}} = \sqrt{32} \cdot \sqrt{x^{6}} = 4\sqrt{2} \cdot x^{3}
   $$
   $$
   \sqrt{98 x^{6}} = \sqrt{98} \cdot \sqrt{x^{6}} = 7\sqrt{2} \cdot x^{3}
   $$

2. **Substituting back:**
   $$
   1.1 \cdot 2 \cdot 4\sqrt{2} \cdot x^{3} + 7\sqrt{2} \cdot x^{3}
   $$

3. **Calculate the first term:**
   $$
   1.1 \cdot 2 \cdot 4 = 8.8 \implies 8.8\sqrt{2} \cdot x^{3}
   $$

4. **Combine the terms:**
   $$
   8.8\sqrt{2} \cdot x^{3} + 7\sqrt{2} \cdot x^{3} = (8.8 + 7)\sqrt{2} \cdot x^{3} = 15.8\sqrt{2} \cdot x^{3}
   $$

Thus, the simplified form of the second expression is:
$$
22.344574x^{3}
$$

### Final Results:
1. The simplified form of the first expression is $$\frac{1}{1250}$$.
2. The simplified form of the second expression is $$22.344574x^{3}$$.
The simplified forms are: 1. $$ \frac{1}{1250} $$ 2. $$ 22.344574x^{3} $$

1.1 Simplify: \( .1 \frac{25^{x+1} \cdot 25^{x-2}}{125^{x+1} \cdot 5^{x-2}} \) 1.1 \( 1.1 \cdot 2 \sqrt{32 x^{6}}+\sqrt{98 x^{6}} \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Algebra Questions