To factor the expression [tex]\( 4x^2 - 25y^6 \)[/tex], follow these detailed steps:
### Step 1: Identify the form of the expression
The given expression is [tex]\( 4x^2 - 25y^6 \)[/tex].
### Step 2: Recognize the difference of perfect squares
Notice that [tex]\( 4x^2 \)[/tex] and [tex]\( 25y^6 \)[/tex] are both perfect squares. Specifically:
- [tex]\( 4x^2 \)[/tex] can be written as [tex]\( (2x)^2 \)[/tex]
- [tex]\( 25y^6 \)[/tex] can be written as [tex]\( (5y^3)^2 \)[/tex]
### Step 3: Use the difference of squares formula
The difference of squares formula states that [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex].
In this case:
- [tex]\( a = 2x \)[/tex]
- [tex]\( b = 5y^3 \)[/tex]
### Step 4: Apply the formula
Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the difference of squares formula:
[tex]\[
4x^2 - 25y^6 = (2x)^2 - (5y^3)^2 = (2x + 5y^3)(2x - 5y^3)
\][/tex]
### Step 5: Write the factored form
Therefore, the factored form of the expression [tex]\( 4x^2 - 25y^6 \)[/tex] is:
[tex]\[
(2x + 5y^3)(2x - 5y^3)
\][/tex]
So, the expression [tex]\( 4x^2 - 25y^6 \)[/tex] factors as [tex]\((2x + 5y^3)(2x - 5y^3)\)[/tex].
