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Sagot :
To determine which multiplication expression is equivalent to [tex]\(\frac{x+8}{x^2} + \frac{2x+16}{2x^2}\)[/tex], let's start by simplifying the original expression step by step.
1. Consider the original expression:
[tex]\[ \frac{x+8}{x^2} + \frac{2x+16}{2x^2} \][/tex]
2. Factor the numerator in the second fraction:
[tex]\[ \frac{2x+16}{2x^2} = \frac{2(x+8)}{2x^2} = \frac{x+8}{x^2} \][/tex]
3. The expression now is:
[tex]\[ \frac{x+8}{x^2} + \frac{x+8}{x^2} \][/tex]
4. Combine the two fractions:
[tex]\[ \frac{(x+8) + (x+8)}{x^2} = \frac{2(x+8)}{x^2} \][/tex]
This further simplifies to:
[tex]\[ 2 \cdot \frac{x+8}{x^2} \][/tex]
5. Next, we want to find the equivalent multiplication expression. We analyze the given options:
a. [tex]\(\frac{x^2}{x+8} \cdot \frac{2 x^2}{2 x+16}\)[/tex]
b. [tex]\(\frac{x+8}{x^2} \cdot \frac{2 x+16}{2 x^2}\)[/tex]
c. [tex]\(\frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16}\)[/tex]
6. For each of these options, let's simplify to see if it matches [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex]:
- For the first option:
[tex]\[ \frac{x^2}{x+8} \cdot \frac{2 x^2}{2 x+16} \][/tex]
Simplifying the second fraction:
[tex]\[ \frac{2 x^2}{2(x+8)} = \frac{x^2}{x+8} \][/tex]
Then, multiplying both fractions:
[tex]\[ \frac{x^2}{x+8} \cdot \frac{x^2}{x+8} = \left(\frac{x^2}{x+8}\right)^2 \][/tex]
This does not simplify to [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex].
- For the second option:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{2 x+16}{2 x^2} \][/tex]
Simplifying the second fraction:
[tex]\[ \frac{2(x+8)}{2x^2} = \frac{x+8}{x^2} \][/tex]
Then, multiplying both fractions:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{x+8}{x^2} = \left(\frac{x+8}{x^2}\right)^2 \][/tex]
This also does not simplify to [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex].
- For the third option:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16} \][/tex]
Simplifying the second fraction:
[tex]\[ \frac{2 x^2}{2(x+8)} = x^2 \cdot \frac{1}{(x+8)} \][/tex]
Then, multiplying both fractions:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{x^2}{x+8} = 1 \][/tex]
Hence:
[tex]\[ 2 \cdot \frac{x+8}{x^2} \][/tex]
This matches the original simplified expression.
Therefore, the correct multiplication expression is:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16} \][/tex]
The correct choice is:
[tex]\[ \boxed{\frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16}} \][/tex]
1. Consider the original expression:
[tex]\[ \frac{x+8}{x^2} + \frac{2x+16}{2x^2} \][/tex]
2. Factor the numerator in the second fraction:
[tex]\[ \frac{2x+16}{2x^2} = \frac{2(x+8)}{2x^2} = \frac{x+8}{x^2} \][/tex]
3. The expression now is:
[tex]\[ \frac{x+8}{x^2} + \frac{x+8}{x^2} \][/tex]
4. Combine the two fractions:
[tex]\[ \frac{(x+8) + (x+8)}{x^2} = \frac{2(x+8)}{x^2} \][/tex]
This further simplifies to:
[tex]\[ 2 \cdot \frac{x+8}{x^2} \][/tex]
5. Next, we want to find the equivalent multiplication expression. We analyze the given options:
a. [tex]\(\frac{x^2}{x+8} \cdot \frac{2 x^2}{2 x+16}\)[/tex]
b. [tex]\(\frac{x+8}{x^2} \cdot \frac{2 x+16}{2 x^2}\)[/tex]
c. [tex]\(\frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16}\)[/tex]
6. For each of these options, let's simplify to see if it matches [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex]:
- For the first option:
[tex]\[ \frac{x^2}{x+8} \cdot \frac{2 x^2}{2 x+16} \][/tex]
Simplifying the second fraction:
[tex]\[ \frac{2 x^2}{2(x+8)} = \frac{x^2}{x+8} \][/tex]
Then, multiplying both fractions:
[tex]\[ \frac{x^2}{x+8} \cdot \frac{x^2}{x+8} = \left(\frac{x^2}{x+8}\right)^2 \][/tex]
This does not simplify to [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex].
- For the second option:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{2 x+16}{2 x^2} \][/tex]
Simplifying the second fraction:
[tex]\[ \frac{2(x+8)}{2x^2} = \frac{x+8}{x^2} \][/tex]
Then, multiplying both fractions:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{x+8}{x^2} = \left(\frac{x+8}{x^2}\right)^2 \][/tex]
This also does not simplify to [tex]\(2 \cdot \frac{x+8}{x^2}\)[/tex].
- For the third option:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16} \][/tex]
Simplifying the second fraction:
[tex]\[ \frac{2 x^2}{2(x+8)} = x^2 \cdot \frac{1}{(x+8)} \][/tex]
Then, multiplying both fractions:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{x^2}{x+8} = 1 \][/tex]
Hence:
[tex]\[ 2 \cdot \frac{x+8}{x^2} \][/tex]
This matches the original simplified expression.
Therefore, the correct multiplication expression is:
[tex]\[ \frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16} \][/tex]
The correct choice is:
[tex]\[ \boxed{\frac{x+8}{x^2} \cdot \frac{2 x^2}{2 x+16}} \][/tex]
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