Find the product `(7a-5b)(49a^(2)+35ab+25b^(2))`.

Answer

To find the product \((7a - 5b)(49a^2 + 35ab + 25b^2)\), we can recognize that the second factor, \(49a^2 + 35ab + 25b^2\), is a perfect square trinomial. It can be expressed as \((7a - 5b)^2\).

Let's break down the steps:

### Step 1: Identify the Perfect Square Trinomial
The expression \(49a^2 + 35ab + 25b^2\) can be rewritten as:
\[ (7a)^2 + 2(7a)(5b) + (5b)^2 \]
This matches the form of \((x + y)^2\) where \(x = 7a\) and \(y = 5b\). Thus, we can write:
\[ 49a^2 + 35ab + 25b^2 = (7a + 5b)^2 \]

### Step 2: Rewrite the Original Expression
Now we can substitute this back into our original expression:
\[ (7a - 5b)(49a^2 + 35ab + 25b^2) = (7a - 5b)(7a + 5b)^2 \]

### Step 3: Use the Difference of Squares Formula
We can use the difference of squares formula, which states that \( (x - y)(x + y) = x^2 - y^2 \). Here, we let \(x = 7a\) and \(y = 5b\):
\[ (7a - 5b)(7a + 5b) = (7a)^2 - (5b)^2 \]
Calculating this gives:
\[ 49a^2 - 25b^2 \]

### Step 4: Combine the Results
Now we can multiply this result by \((7a + 5b)\):
\[ (49a^2 - 25b^2)(7a + 5b) \]

### Step 5: Expand the Expression
Now we can expand this expression using the distributive property (FOIL method):
1. \(49a^2 \cdot 7a = 343a^3\)
2. \(49a^2 \cdot 5b = 245a^2b\)
3. \(-25b^2 \cdot 7a = -175ab^2\)
4. \(-25b^2 \cdot 5b = -125b^3\)

Combining all these terms, we get:
\[ 343a^3 + 245a^2b - 175ab^2 - 125b^3 \]

### Final Answer
Thus, the product \((7a - 5b)(49a^2 + 35ab + 25b^2)\) is:
\[ 343a^3 + 245a^2b - 175ab^2 - 125b^3 \]