- por + da -
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los exponente:

Ejercicio 1

1. [tex]\((a^x - a^{x+1} + a^{x+2}) \cdot (a + 1)\)[/tex]
Continue [tex]$\square$[/tex]
2. [tex]\((x^{n+1} + 2x^{n+2} - x^{n+3}) \cdot (x^2 + x)\)[/tex]
3. [tex]\((m^{a-1} + m^{a+1} + m^{a+2} - m^a) \cdot (m^2 - 2m + 3)\)[/tex]
4. [tex]\((a^{n+2} - 2a^n + 3a^{n+1}) \cdot (a^n + a^{n+1})\)[/tex]
5. [tex]\((x^{a+2} - x^a + 2x^{a+1}) \cdot (x^{a+3} - 2x^{a+1})\)[/tex]
6. [tex]\((3a^{x-2} - 2a^{x-1} + a^x) \cdot (a^2 + 2a - 1)\)[/tex]
7. [tex]\((3a^{x-1} + a^x - 2a^{x-2}) \cdot (a^x - a^{x-1} + a^{x-2})\)[/tex]
8. [tex]\((m^{a-1} - 2m^{a+2} - m^{a+3} + m^{a+4}) \cdot (m^{a-3} - m^{a-1} + m^{a-2})\)[/tex]
9. [tex]\((x^{a-1} + 2x^{a-2} - x^{a-3} + x^{a-4}) \cdot (-x^{a-3} + x^{a-1} - x^{a-2})\)[/tex]
10. [tex]\((a^n b - a^{n-1} b^2 + 2a^{n-2} b^3 - a^{n-3} b^4) \cdot (a^n b^2 - a^{n-2} b^4)\)[/tex]

6. [tex]\(\frac{3}{8}\)[/tex]
7. [tex]\(\frac{1}{3}\)[/tex]
8. [tex]\(\frac{2}{7}\)[/tex]
9. [tex]\(\frac{1}{2}\)[/tex]



Answer :

Let's tackle the specific problem you mentioned:

Exercise 1:

1. [tex]\(a^x - a^{x+1} + a^{x+2}\)[/tex] multiplied by [tex]\(a + 1\)[/tex]

To solve this, let's go through the steps one by one.

### Step-by-Step Solution

#### Step 1: Write Down the Given Expression

The expression given is:
[tex]\[a^x - a^{x+1} + a^{x+2}\][/tex]

#### Step 2: Identify the Expression to Multiply By

We need to multiply this expression by:
[tex]\[a + 1\][/tex]

#### Step 3: Multiply the Expressions

Multiply the expressions together:
[tex]\[(a^x - a^{x+1} + a^{x+2})(a + 1)\][/tex]

#### Step 4: Distribute the Terms

To distribute:
[tex]\[ (a^x - a^{x+1} + a^{x+2})(a + 1) = a^x \cdot (a + 1) - a^{x+1} \cdot (a + 1) + a^{x+2} \cdot (a + 1) \][/tex]

Distributing each term individually:

1. [tex]\(a^x \cdot (a + 1)\)[/tex]
- [tex]\(a^x \cdot a = a^{x+1}\)[/tex]
- [tex]\(a^x \cdot 1 = a^x\)[/tex]
[tex]\[a^x(a + 1) = a^{x+1} + a^x\][/tex]

2. [tex]\(-a^{x+1} \cdot (a + 1)\)[/tex]
- [tex]\(-a^{x+1} \cdot a = -a^{x+2}\)[/tex]
- [tex]\(-a^{x+1} \cdot 1 = -a^{x+1}\)[/tex]
[tex]\[-a^{x+1}(a + 1) = -a^{x+2} - a^{x+1}\][/tex]

3. [tex]\(a^{x+2} \cdot (a + 1)\)[/tex]
- [tex]\(a^{x+2} \cdot a = a^{x+3}\)[/tex]
- [tex]\(a^{x+2} \cdot 1 = a^{x+2}\)[/tex]
[tex]\[a^{x+2}(a + 1) = a^{x+3} + a^{x+2}\][/tex]

#### Step 5: Combine the Terms

Finally, combine all the distributed terms:
[tex]\[ a^{x+1} + a^x - a^{x+2} - a^{x+1} + a^{x+3} + a^{x+2} \][/tex]

Simplify by combining like terms:
[tex]\[ a^x - a^{x+1} + a^{x+2} + a^x = (a + 1)(a^x - a^{x+1} + a^{x+2}) \][/tex]

Thus, the resulting expression is:
[tex]\[ (a + 1)(a^x - a^{x+1} + a^{x+2}) \][/tex]

And there you have it! We have successfully multiplied the given expression by [tex]\(a + 1\)[/tex] and arrived at the result:
[tex]\[ (a + 1)(a^x - a^{x+1} + a^{x+2}) \][/tex]