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Sagot :
Let's simplify the given expression step-by-step to find:
[tex]\[ \left(\frac{x^a}{x^{-b}}\right)^{a-b} \times \left(\frac{x^b}{x^c}\right)^{b+c} \times \left(\frac{x^{-a}}{x^{-c}}\right)^{c+a} \][/tex]
### Step 1: Simplify each of the individual fractions inside the parentheses
#### First Term: [tex]\(\left(\frac{x^a}{x^{-b}}\right)^{a-b}\)[/tex]
Using the property of exponents [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex], we get:
[tex]\[ \frac{x^a}{x^{-b}} = x^{a - (-b)} = x^{a + b} \][/tex]
So the first term is:
[tex]\[ \left(x^{a + b}\right)^{a - b} \][/tex]
#### Second Term: [tex]\(\left(\frac{x^b}{x^c}\right)^{b+c}\)[/tex]
Similarly,
[tex]\[ \frac{x^b}{x^c} = x^{b - c} \][/tex]
So the second term is:
[tex]\[ \left(x^{b - c}\right)^{b + c} \][/tex]
#### Third Term: [tex]\(\left(\frac{x^{-a}}{x^{-c}}\right)^{c+a}\)[/tex]
Again, applying the same property,
[tex]\[ \frac{x^{-a}}{x^{-c}} = x^{-a - (-c)} = x^{-a + c} \][/tex]
So the third term is:
[tex]\[ \left(x^{-a + c}\right)^{c + a} \][/tex]
### Step 2: Apply the Power Rule [tex]\((x^m)^n = x^{mn}\)[/tex] to each term
Now we will apply the power of a power rule to each of the simplified terms:
1. [tex]\(\left(x^{a + b}\right)^{a - b} = x^{(a + b)(a - b)}\)[/tex]
2. [tex]\(\left(x^{b - c}\right)^{b + c} = x^{(b - c)(b + c)}\)[/tex]
3. [tex]\(\left(x^{-a + c}\right)^{c + a} = x^{(-a + c)(c + a)}\)[/tex]
### Step 3: Multiply the Simplified Expressions
Our expression now is:
[tex]\[ x^{(a + b)(a - b)} \times x^{(b - c)(b + c)} \times x^{(-a + c)(c + a)} \][/tex]
We can combine these as:
[tex]\[ x^{(a + b)(a - b) + (b - c)(b + c) + (-a + c)(c + a)} \][/tex]
### Step 4: Simplify the Exponents
Each term inside the exponents can be expanded using the difference of squares formula [tex]\( (m+n)(m-n) = m^2 - n^2 \)[/tex]:
1. [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]
2. [tex]\((b - c)(b + c) = b^2 - c^2\)[/tex]
3. [tex]\((-a + c)(c + a) = c^2 - a^2\)[/tex]
So the entire exponent simplifies as:
[tex]\[ a^2 - b^2 + b^2 - c^2 + c^2 - a^2 \][/tex]
Notice that [tex]\(a^2 - a^2 + b^2 - b^2 + c^2 - c^2 = 0\)[/tex].
### Conclusion
Thus, the overall exponent is zero, meaning our multiplication results in:
[tex]\[ x^0 = 1 \][/tex]
And therefore, the simplified expression is:
[tex]\[ 1 \][/tex]
Final result:
[tex]\[ (x^{(-a + c)})^{c + a} \cdot (x^{a + b})^{a - b} \cdot (x^{b - c})^{b + c} = 1 \][/tex]
[tex]\[ \left(\frac{x^a}{x^{-b}}\right)^{a-b} \times \left(\frac{x^b}{x^c}\right)^{b+c} \times \left(\frac{x^{-a}}{x^{-c}}\right)^{c+a} \][/tex]
### Step 1: Simplify each of the individual fractions inside the parentheses
#### First Term: [tex]\(\left(\frac{x^a}{x^{-b}}\right)^{a-b}\)[/tex]
Using the property of exponents [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex], we get:
[tex]\[ \frac{x^a}{x^{-b}} = x^{a - (-b)} = x^{a + b} \][/tex]
So the first term is:
[tex]\[ \left(x^{a + b}\right)^{a - b} \][/tex]
#### Second Term: [tex]\(\left(\frac{x^b}{x^c}\right)^{b+c}\)[/tex]
Similarly,
[tex]\[ \frac{x^b}{x^c} = x^{b - c} \][/tex]
So the second term is:
[tex]\[ \left(x^{b - c}\right)^{b + c} \][/tex]
#### Third Term: [tex]\(\left(\frac{x^{-a}}{x^{-c}}\right)^{c+a}\)[/tex]
Again, applying the same property,
[tex]\[ \frac{x^{-a}}{x^{-c}} = x^{-a - (-c)} = x^{-a + c} \][/tex]
So the third term is:
[tex]\[ \left(x^{-a + c}\right)^{c + a} \][/tex]
### Step 2: Apply the Power Rule [tex]\((x^m)^n = x^{mn}\)[/tex] to each term
Now we will apply the power of a power rule to each of the simplified terms:
1. [tex]\(\left(x^{a + b}\right)^{a - b} = x^{(a + b)(a - b)}\)[/tex]
2. [tex]\(\left(x^{b - c}\right)^{b + c} = x^{(b - c)(b + c)}\)[/tex]
3. [tex]\(\left(x^{-a + c}\right)^{c + a} = x^{(-a + c)(c + a)}\)[/tex]
### Step 3: Multiply the Simplified Expressions
Our expression now is:
[tex]\[ x^{(a + b)(a - b)} \times x^{(b - c)(b + c)} \times x^{(-a + c)(c + a)} \][/tex]
We can combine these as:
[tex]\[ x^{(a + b)(a - b) + (b - c)(b + c) + (-a + c)(c + a)} \][/tex]
### Step 4: Simplify the Exponents
Each term inside the exponents can be expanded using the difference of squares formula [tex]\( (m+n)(m-n) = m^2 - n^2 \)[/tex]:
1. [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]
2. [tex]\((b - c)(b + c) = b^2 - c^2\)[/tex]
3. [tex]\((-a + c)(c + a) = c^2 - a^2\)[/tex]
So the entire exponent simplifies as:
[tex]\[ a^2 - b^2 + b^2 - c^2 + c^2 - a^2 \][/tex]
Notice that [tex]\(a^2 - a^2 + b^2 - b^2 + c^2 - c^2 = 0\)[/tex].
### Conclusion
Thus, the overall exponent is zero, meaning our multiplication results in:
[tex]\[ x^0 = 1 \][/tex]
And therefore, the simplified expression is:
[tex]\[ 1 \][/tex]
Final result:
[tex]\[ (x^{(-a + c)})^{c + a} \cdot (x^{a + b})^{a - b} \cdot (x^{b - c})^{b + c} = 1 \][/tex]
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