Question

Factor the expression completely: 81x⁴ - 625y⁴. Simplify your answer.

Answer 1

The expression is a difference of squares, and can be factored out completely using the formula
`(a+b)(a-b)=a^2-b^2`

Notice that

`9^2=81\\25^2=625\\`

and
`(x^2)^2=x^4` because
`(a^x)(a^y) =a^x^+^y` by the product rule of exponents

`81x^4 - 625y^4\\`

Using the product property, we can simply this expression into

`(9)^2(x^2)^2 - (25)^2(y^2)^2`

Therefore, the factored form is

`(9x^2+25y^2)(9x^2-25y^2)`

Answer 2

To factor the expression completely, we can use the difference of squares formula and then further factor if necessary.

Step-by-step explanation:

To factor the expression 81x⁴ - 625y⁴ completely, we can use the difference of squares formula. The formula states that a² - b² can be factored as (a + b)(a - b). In this case, 81x⁴ is the square of 9x² and 625y⁴ is the square of 25y². Therefore, we have:

81x⁴ - 625y⁴ = (9x² + 25y²)(9x² - 25y²)

To simplify the answer, we can further factor the difference of squares 9x² - 25y² as (3x + 5y)(3x - 5y), which gives us the completely factored expression:

81x⁴ - 625y⁴ = (9x² + 25y²)(3x + 5y)(3x - 5y)