Question

1. factor \\(25a^2 - 81b^2\\).

\\((5a - 9b)(5a - 9b)\\)

\\((12.5a - 4.5b)(12.5a + 4.5b)\\)

\\((5a + 9b)(5a + 9b)\\)

\\((5a - 9b)(5a + 9b)\\)

Explanation:

Identify the expression structure

Using the Exponent Rules and Algebraic Simplification knowledge points

$$ 25a^2 - 81b^2 = (5a)^2 - (9b)^2 $$

Apply the difference of squares formula

We apply the difference of squares identity \(x^2 - y^2 = (x - y)(x + y)\) where \(x = 5a\) and \(y = 9b\):

$$ (5a)^2 - (9b)^2 = (5a - 9b)(5a + 9b) $$

Match with the given options

We compare our factored expression with the choices:

• \((5a - 9b)(5a - 9b)\) is a perfect square trinomial factor.
• \((12.5a - 4.5b)(12.5a + 4.5b)\) incorrectly divides the coefficients by 2 instead of taking square roots.
• \((5a + 9b)(5a + 9b)\) is a perfect square trinomial factor.
• \((5a - 9b)(5a + 9b)\) matches our result.

Answer:

• (A) \((5a - 9b)(5a - 9b)\)
• (B) \((12.5a - 4.5b)(12.5a + 4.5b)\)
• (C) \((5a + 9b)(5a + 9b)\)
• (D) \((5a - 9b)(5a + 9b)\) (Correct answer)