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]
