Which is equivalent to (9y² - 4x)(9y² + 4x), and what type of special product is it?
Understand the Problem
The question is asking to determine what expression is equivalent to (9y² - 4x)(9y² + 4x) and identify the type of special product it represents. This involves applying the difference of squares formula.
Answer
The expression is $81y^4 - 16x²$ and it is the difference of squares.
Answer for screen readers
The expression equivalent to $(9y² - 4x)(9y² + 4x)$ is $81y^4 - 16x²$, and it represents the difference of squares.
Steps to Solve
- Identify the Expression Type
We start with the expression $(9y² - 4x)(9y² + 4x)$. This matches the form of a difference of squares: $(a - b)(a + b) = a² - b²$, where $a = 9y²$ and $b = 4x$.
- Apply the Difference of Squares Formula
Using the difference of squares formula, we can rewrite the expression:
$$ (9y² - 4x)(9y² + 4x) = (9y²)² - (4x)² $$
- Calculate the Squares
Calculate each square:
-
Calculate $(9y²)²$:
- This results in $81y^4$.
-
Calculate $(4x)²$:
- This results in $16x²$.
Putting this together gives:
$$ 81y^4 - 16x² $$
- Determine the Product Type
The resulting expression $81y^4 - 16x²$ represents the difference of squares.
The expression equivalent to $(9y² - 4x)(9y² + 4x)$ is $81y^4 - 16x²$, and it represents the difference of squares.
More Information
The difference of squares is a special product that occurs when we multiply two binomials that are structured as $(a - b)(a + b)$. This concept is essential in algebra as it helps simplify expressions and solve equations.
Tips
- Confusing the difference of squares with a perfect square trinomial. A perfect square trinomial takes the form $(a + b)² = a² + 2ab + b²$, whereas a difference of squares involves subtraction.
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