Solve ax^4-81ay^4

$$a { x }^{ 4 } -81a { y }^{ 4 }$$

Factor

$$a\left(x-3y\right)\left(x+3y\right)\left(x^{2}+9y^{2}\right)$$

Solution Steps

Factor out a.

$$a\left(x^{4}-81y^{4}\right)$$

Consider x^{4}-81y^{4}. Rewrite x^{4}-81y^{4} as \left(x^{2}\right)^{2}-\left(9y^{2}\right)^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(x^{2}-9y^{2}\right)\left(x^{2}+9y^{2}\right)$$

Consider x^{2}-9y^{2}. Rewrite x^{2}-9y^{2} as x^{2}-\left(3y\right)^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(x-3y\right)\left(x+3y\right)$$

Rewrite the complete factored expression.

$$a\left(x-3y\right)\left(x+3y\right)\left(x^{2}+9y^{2}\right)$$

Evaluate

$$a\left(x^{4}-81y^{4}\right)$$