1. Simplifica las siguientes expresiones. Explica el procedimiento que realizaste. a. $$ \frac{3}{7}+3,6+\frac{4}{9}-2,9+14 $$ b. $$ \sqrt{6}+7 \sqrt{6}-4-3 \sqrt{6}+\frac{1}{2} $$ c. $$ \frac{\sqrt{3}}{3}-\sqrt{3}-\left(5 \sqrt{3}-\frac{\sqrt{3}}{4}\right) $$ d. $$ \left(\sqrt{2}+\sqrt{7}-\frac{3}{8} \sqrt{2}\right)+\sqrt{11}-8 \sqrt{7}+2 $$ e. $$ 2, \overline{8}+6, \overline{4}+12, \overline{4}-9,4 \overline{3} $$ f. $$ 0, \overline{20}+0, \overline{02}+0, \overline{002}-2, \overline{222} $$

Question

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$0.2+0.02+0.002-2.222$$
- step1: Calculate:
  $$-2$$
Calculate or simplify the expression $$ 3/7 + 3.6 + 4/9 - 2.9 + 14 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3}{7}+3.6+\frac{4}{9}-2.9+14$$
- step1: Calculate:
  $$\frac{3}{7}+14.7+\frac{4}{9}$$
- step2: Rewrite the expression:
  $$\frac{3}{7}+\frac{147}{10}+\frac{4}{9}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{3\times 10\times 9}{7\times 10\times 9}+\frac{147\times 7\times 9}{10\times 7\times 9}+\frac{4\times 7\times 10}{9\times 7\times 10}$$
- step4: Multiply the terms:
  $$\frac{3\times 10\times 9}{630}+\frac{147\times 7\times 9}{10\times 7\times 9}+\frac{4\times 7\times 10}{9\times 7\times 10}$$
- step5: Multiply the terms:
  $$\frac{3\times 10\times 9}{630}+\frac{147\times 7\times 9}{630}+\frac{4\times 7\times 10}{9\times 7\times 10}$$
- step6: Multiply the terms:
  $$\frac{3\times 10\times 9}{630}+\frac{147\times 7\times 9}{630}+\frac{4\times 7\times 10}{630}$$
- step7: Transform the expression:
  $$\frac{3\times 10\times 9+147\times 7\times 9+4\times 7\times 10}{630}$$
- step8: Multiply the terms:
  $$\frac{270+147\times 7\times 9+4\times 7\times 10}{630}$$
- step9: Multiply the terms:
  $$\frac{270+9261+4\times 7\times 10}{630}$$
- step10: Multiply the terms:
  $$\frac{270+9261+280}{630}$$
- step11: Add the numbers:
  $$\frac{9811}{630}$$
Calculate or simplify the expression $$ \sqrt(3)/3 - \sqrt(3) - (5*\sqrt(3) - \sqrt(3)/4) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\sqrt{3}}{3}-\sqrt{3}-\left(5\sqrt{3}-\frac{\sqrt{3}}{4}\right)$$
- step1: Subtract the numbers:
  $$\frac{\sqrt{3}}{3}-\sqrt{3}-\frac{19\sqrt{3}}{4}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{\sqrt{3}\times 4}{3\times 4}-\frac{\sqrt{3}\times 3\times 4}{3\times 4}-\frac{19\sqrt{3}\times 3}{4\times 3}$$
- step3: Multiply the numbers:
  $$\frac{\sqrt{3}\times 4}{12}-\frac{\sqrt{3}\times 3\times 4}{3\times 4}-\frac{19\sqrt{3}\times 3}{4\times 3}$$
- step4: Multiply the numbers:
  $$\frac{\sqrt{3}\times 4}{12}-\frac{\sqrt{3}\times 3\times 4}{12}-\frac{19\sqrt{3}\times 3}{4\times 3}$$
- step5: Multiply the numbers:
  $$\frac{\sqrt{3}\times 4}{12}-\frac{\sqrt{3}\times 3\times 4}{12}-\frac{19\sqrt{3}\times 3}{12}$$
- step6: Transform the expression:
  $$\frac{\sqrt{3}\times 4-\sqrt{3}\times 3\times 4-19\sqrt{3}\times 3}{12}$$
- step7: Reorder the terms:
  $$\frac{4\sqrt{3}-\sqrt{3}\times 3\times 4-19\sqrt{3}\times 3}{12}$$
- step8: Multiply the terms:
  $$\frac{4\sqrt{3}-12\sqrt{3}-19\sqrt{3}\times 3}{12}$$
- step9: Multiply the terms:
  $$\frac{4\sqrt{3}-12\sqrt{3}-57\sqrt{3}}{12}$$
- step10: Subtract the numbers:
  $$\frac{-65\sqrt{3}}{12}$$
- step11: Rewrite the fraction:
  $$-\frac{65\sqrt{3}}{12}$$
Calculate or simplify the expression $$ \sqrt(6) + 7*\sqrt(6) - 4 - 3*\sqrt(6) + 1/2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{6}+7\sqrt{6}-4-3\sqrt{6}+\frac{1}{2}$$
- step1: Calculate:
  $$5\sqrt{6}-4+\frac{1}{2}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{5\sqrt{6}\times 2}{2}-\frac{4\times 2}{2}+\frac{1}{2}$$
- step3: Transform the expression:
  $$\frac{5\sqrt{6}\times 2-4\times 2+1}{2}$$
- step4: Multiply the terms:
  $$\frac{10\sqrt{6}-4\times 2+1}{2}$$
- step5: Multiply the numbers:
  $$\frac{10\sqrt{6}-8+1}{2}$$
- step6: Add the numbers:
  $$\frac{10\sqrt{6}-7}{2}$$
Calculate or simplify the expression $$ 2.8 + 6.4 + 12.4 - 9.43 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2.8+6.4+12.4-9.43$$
- step1: Calculate:
  $$12.17$$
Calculate or simplify the expression $$ (\sqrt(2) + \sqrt(7) - (3/8)*\sqrt(2)) + \sqrt(11) - 8*\sqrt(7) + 2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\sqrt{2}+\sqrt{7}-\frac{3}{8}\sqrt{2}\right)+\sqrt{11}-8\sqrt{7}+2$$
- step1: Multiply:
  $$\left(\sqrt{2}+\sqrt{7}-\frac{3\sqrt{2}}{8}\right)+\sqrt{11}-8\sqrt{7}+2$$
- step2: Calculate:
  $$\frac{5\sqrt{2}+8\sqrt{7}}{8}+\sqrt{11}-8\sqrt{7}+2$$
- step3: Reduce fractions to a common denominator:
  $$\frac{5\sqrt{2}+8\sqrt{7}}{8}+\frac{\sqrt{11}\times 8}{8}-\frac{8\sqrt{7}\times 8}{8}+\frac{2\times 8}{8}$$
- step4: Transform the expression:
  $$\frac{5\sqrt{2}+8\sqrt{7}+\sqrt{11}\times 8-8\sqrt{7}\times 8+2\times 8}{8}$$
- step5: Reorder the terms:
  $$\frac{5\sqrt{2}+8\sqrt{7}+8\sqrt{11}-8\sqrt{7}\times 8+2\times 8}{8}$$
- step6: Multiply the terms:
  $$\frac{5\sqrt{2}+8\sqrt{7}+8\sqrt{11}-64\sqrt{7}+2\times 8}{8}$$
- step7: Multiply the numbers:
  $$\frac{5\sqrt{2}+8\sqrt{7}+8\sqrt{11}-64\sqrt{7}+16}{8}$$
- step8: Subtract the numbers:
  $$\frac{5\sqrt{2}-56\sqrt{7}+8\sqrt{11}+16}{8}$$
Aquí están las simplificaciones de las expresiones solicitadas, junto con el procedimiento que seguí para cada una:

### a. $$ \frac{3}{7}+3,6+\frac{4}{9}-2,9+14 $$

1. **Convertir a un denominador común**: Para sumar y restar fracciones, se necesita un denominador común. En este caso, el denominador común de $$7$$ y $$9$$ es $$63$$.
2. **Convertir las fracciones**:
   - $$ \frac{3}{7} = \frac{27}{63} $$
   - $$ \frac{4}{9} = \frac{28}{63} $$
3. **Sustituir y simplificar**:
   $$
   \frac{27}{63} + 3.6 + \frac{28}{63} - 2.9 + 14 = \frac{27 + 28}{63} + 3.6 - 2.9 + 14 = \frac{55}{63} + 3.6 - 2.9 + 14
   $$
4. **Calcular el resultado**:
   $$
   3.6 - 2.9 = 0.7 \quad \text{y} \quad 0.7 + 14 = 14.7
   $$
5. **Resultado final**:
   $$
   14.7 + \frac{55}{63} \approx 15.5\dot{7}3015\dot{8}
   $$

### b. $$ \sqrt{6}+7 \sqrt{6}-4-3 \sqrt{6}+\frac{1}{2} $$

1. **Combinar términos semejantes**:
   $$
   (1 + 7 - 3)\sqrt{6} - 4 + \frac{1}{2} = 5\sqrt{6} - 4 + 0.5
   $$
2. **Simplificar**:
   $$
   -4 + 0.5 = -3.5
   $$
3. **Resultado final**:
   $$
   5\sqrt{6} - 3.5 = \frac{10\sqrt{6}-7}{2}
   $$

### c. $$ \frac{\sqrt{3}}{3}-\sqrt{3}-\left(5 \sqrt{3}-\frac{\sqrt{3}}{4}\right) $$

1. **Expandir la expresión**:
   $$
   \frac{\sqrt{3}}{3} - \sqrt{3} - 5\sqrt{3} + \frac{\sqrt{3}}{4}
   $$
2. **Combinar términos semejantes**:
   $$
   \left(\frac{1}{3} + \frac{1}{4} - 1 - 5\right)\sqrt{3} = \left(\frac{4 + 3 - 12 - 60}{12}\right)\sqrt{3} = -\frac{65\sqrt{3}}{12}
   $$
3. **Resultado final**:
   $$
   -\frac{65\sqrt{3}}{12}
   $$

### d. $$ \left(\sqrt{2}+\sqrt{7}-\frac{3}{8} \sqrt{2}\right)+\sqrt{11}-8 \sqrt{7}+2 $$

1. **Combinar términos semejantes**:
   $$
   \left(1 - \frac{3}{8}\right)\sqrt{2} + (1 - 8)\sqrt{7} + \sqrt{11} + 2
   $$
2. **Simplificar**:
   $$
   \frac{5}{8}\sqrt{2} - 7\sqrt{7} + \sqrt{11} + 2
   $$
3. **Resultado final**:
   $$
   \frac{5\sqrt{2}-56\sqrt{7}+8\sqrt{11}+16}{8}
   $$

### e. $$ 2, \overline{8}+6, \overline{4}+12, \overline{4}-9,4 \overline{3} $$

1. **Convertir a fracciones**:
   - $$ 2.\overline{8} = \frac{28}{9} $$
   - $$ 6.\overline{4} = \frac{64}{9} $$
   - $$ 12.\overline{4} = \frac{124}{9} $$
   - $$ 9.4\overline{3} = \frac{94}{9} $$
2. **Sumar y restar**:
   $$
   \frac{28 + 64 + 124 - 94}{9} = \frac{122}{9} \approx 12.17
   $$
3. **Resultado final**:
   $$
   12.17
   $$

### f. $$ 0, \overline{20}+0, \overline{02}+0, \overline{002}-2, \overline{222} $$

1. **Convertir a fracciones**:
   - $$ 0.\overline{20} = \frac{20}{99} $$
   - $$ 0.\overline{02} = \frac{2}{99} $$
   - $$ 0.\overline{002} = \frac{2}{990} = \frac{1}{495} $$
   - $$ 2.\overline{222} = \frac{222}{99} $$
2. **Sumar y restar**:
   $$
   \frac{20 + 2 + \frac{2}{10} - 222}{99} = \frac{20 + 2 + 0.02 - 222}{99} = \frac{-200}{99} \approx -2
   $$
3. **Resultado final**:
   $$
   -2
   $$

Si necesitas más detalles sobre algún paso específico, no dudes en preguntar.
a. $$ \frac{3}{7} + 3.6 + \frac{4}{9} - 2.9 + 14 = 15.5\dot{7}3015\dot{8} $$ b. $$ \sqrt{6} + 7\sqrt{6} - 4 - 3\sqrt{6} + \frac{1}{2} = 5\sqrt{6} - 3.5 $$ c. $$ \frac{\sqrt{3}}{3} - \sqrt{3} - (5\sqrt{3} - \frac{\sqrt{3}}{4}) = -\frac{65\sqrt{3}}{12} $$ d. $$ (\sqrt{2} + \sqrt{7} - \frac{3}{8}\sqrt{2}) + \sqrt{11} - 8\sqrt{7} + 2 = \frac{5\sqrt{2} - 56\sqrt{7} + 8\sqrt{11} + 16}{8} $$ e. $$ 2.\overline{8} + 6.\overline{4} + 12.\overline{4} - 9.4\overline{3} = 12.17 $$ f. $$ 0.\overline{20} + 0.\overline{02} + 0.\overline{002} - 2.\overline{222} = -2 $$

Simplifica las siguientes expresiones.
Explica el procedimiento que realizaste.
a.
b.
c.
d.
e.
f.

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Solución

a.

b.

c.

d.

e.

f.

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