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Sagot :
To solve the expression [tex]\((7m - 5n)^3 - (7m + 5n)^3\)[/tex], let's approach it step by step using algebraic manipulation and known identities.
### Step 1: Recognize the Cubes Difference Identity
The expression [tex]\((a)^3 - (b)^3\)[/tex] can be factored using the difference of cubes identity:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
where [tex]\(a = 7m - 5n\)[/tex] and [tex]\(b = 7m + 5n\)[/tex].
### Step 2: Apply the Identity
Substitute [tex]\(a = 7m - 5n\)[/tex] and [tex]\(b = 7m + 5n\)[/tex] into the difference of cubes identity:
[tex]\[ (7m - 5n)^3 - (7m + 5n)^3 = ((7m - 5n) - (7m + 5n))((7m - 5n)^2 + (7m - 5n)(7m + 5n) + (7m + 5n)^2) \][/tex]
### Step 3: Simplify the First Term
Calculate [tex]\((7m - 5n) - (7m + 5n)\)[/tex]:
[tex]\[ (7m - 5n) - (7m + 5n) = 7m - 5n - 7m - 5n = -10n \][/tex]
### Step 4: Simplify the Second Term
Next, we need to simplify the expression inside the parentheses:
[tex]\[ (7m - 5n)^2 + (7m - 5n)(7m + 5n) + (7m + 5n)^2 \][/tex]
Calculate each part separately:
1. [tex]\((7m - 5n)^2\)[/tex]:
[tex]\[ (7m - 5n)^2 = (7m)^2 - 2 \cdot 7m \cdot 5n + (5n)^2 = 49m^2 - 70mn + 25n^2 \][/tex]
2. [tex]\((7m - 5n)(7m + 5n)\)[/tex]:
[tex]\[ (7m - 5n)(7m + 5n) = (7m)^2 - (5n)^2 = 49m^2 - 25n^2 \][/tex]
3. [tex]\((7m + 5n)^2\)[/tex]:
[tex]\[ (7m + 5n)^2 = (7m)^2 + 2 \cdot 7m \cdot 5n + (5n)^2 = 49m^2 + 70mn + 25n^2 \][/tex]
### Step 5: Combine the Parts
Add the three parts together:
[tex]\[ (49m^2 - 70mn + 25n^2) + (49m^2 - 25n^2) + (49m^2 + 70mn + 25n^2) \][/tex]
Combine like terms:
[tex]\[ 49m^2 - 70mn + 25n^2 + 49m^2 - 25n^2 + 49m^2 + 70mn + 25n^2 = 147m^2 + 25n^2 \][/tex]
### Step 6: Put All Together
So, the simplified form of the expression is:
[tex]\[ (7m - 5n)^3 - (7m + 5n)^3 = -10n (147m^2 + 25n^2) \][/tex]
### Final Simplified Form
This yields the final expression as:
[tex]\[ -1470m^2n - 250n^3 \][/tex]
Thus, the fully simplified result for [tex]\((7m - 5n)^3 - (7m + 5n)^3\)[/tex] is:
[tex]\[ -1470m^2n - 250n^3 \][/tex]
### Step 1: Recognize the Cubes Difference Identity
The expression [tex]\((a)^3 - (b)^3\)[/tex] can be factored using the difference of cubes identity:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
where [tex]\(a = 7m - 5n\)[/tex] and [tex]\(b = 7m + 5n\)[/tex].
### Step 2: Apply the Identity
Substitute [tex]\(a = 7m - 5n\)[/tex] and [tex]\(b = 7m + 5n\)[/tex] into the difference of cubes identity:
[tex]\[ (7m - 5n)^3 - (7m + 5n)^3 = ((7m - 5n) - (7m + 5n))((7m - 5n)^2 + (7m - 5n)(7m + 5n) + (7m + 5n)^2) \][/tex]
### Step 3: Simplify the First Term
Calculate [tex]\((7m - 5n) - (7m + 5n)\)[/tex]:
[tex]\[ (7m - 5n) - (7m + 5n) = 7m - 5n - 7m - 5n = -10n \][/tex]
### Step 4: Simplify the Second Term
Next, we need to simplify the expression inside the parentheses:
[tex]\[ (7m - 5n)^2 + (7m - 5n)(7m + 5n) + (7m + 5n)^2 \][/tex]
Calculate each part separately:
1. [tex]\((7m - 5n)^2\)[/tex]:
[tex]\[ (7m - 5n)^2 = (7m)^2 - 2 \cdot 7m \cdot 5n + (5n)^2 = 49m^2 - 70mn + 25n^2 \][/tex]
2. [tex]\((7m - 5n)(7m + 5n)\)[/tex]:
[tex]\[ (7m - 5n)(7m + 5n) = (7m)^2 - (5n)^2 = 49m^2 - 25n^2 \][/tex]
3. [tex]\((7m + 5n)^2\)[/tex]:
[tex]\[ (7m + 5n)^2 = (7m)^2 + 2 \cdot 7m \cdot 5n + (5n)^2 = 49m^2 + 70mn + 25n^2 \][/tex]
### Step 5: Combine the Parts
Add the three parts together:
[tex]\[ (49m^2 - 70mn + 25n^2) + (49m^2 - 25n^2) + (49m^2 + 70mn + 25n^2) \][/tex]
Combine like terms:
[tex]\[ 49m^2 - 70mn + 25n^2 + 49m^2 - 25n^2 + 49m^2 + 70mn + 25n^2 = 147m^2 + 25n^2 \][/tex]
### Step 6: Put All Together
So, the simplified form of the expression is:
[tex]\[ (7m - 5n)^3 - (7m + 5n)^3 = -10n (147m^2 + 25n^2) \][/tex]
### Final Simplified Form
This yields the final expression as:
[tex]\[ -1470m^2n - 250n^3 \][/tex]
Thus, the fully simplified result for [tex]\((7m - 5n)^3 - (7m + 5n)^3\)[/tex] is:
[tex]\[ -1470m^2n - 250n^3 \][/tex]
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