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Question: How do you factor \(16{x^4} - 81{y^4}\)?...

How do you factor 16x4−81y416{x^4} - 81{y^4}?

Explanation

Solution

In this question we need to find the factor of algebraic expression 16x4−81y416{x^4} - 81{y^4}. Given algebraic expression is of two variables xx and yy. To solve this question we need to use the following basic algebraic identities such as a2−b2=(a+b)(a−b){a^2} - {b^2} = (a + b)(a - b). To solve this question we need to know the square root of a number or how to find the square root of a number. To solve this we also need to know the laws of exponents.

Complete step by step solution:
Let us try to solve this question in which we are asked to find the factor of the given algebraic expression 16x4−81y416{x^4} - 81{y^4}. To find the factors of the equation we manipulate the given algebraic expression by using our knowledge of exponents, so that we can apply the algebraic identity a2−b2=(a+b)(a−b){a^2} - {b^2} = (a + b)(a - b). So, let’s come back to the question.
We have to find factor of 16x4−81y416{x^4} - 81{y^4}, this can be written as
16x4−81y4=(4x2)2−(9y2)216{x^4} - 81{y^4} = {(4{x^2})^2} - {(9{y^2})^2} (1)(1)
Because we know that from law of exponents ab⋅c=(ab)c{a^{b \cdot c}} = {({a^b})^c} and also we know that 16=4216 = {4^2} and 81=9281 = {9^2}
Now, applying the identity a2−b2=(a+b)(a−b){a^2} - {b^2} = (a + b)(a - b) in equation (1)(1), we get
(4x2)2−(9x2)2=(4x2−9x2)(4x2+9y2){(4{x^2})^2} - {(9{x^2})^2} = (4{x^2} - 9{x^2})(4{x^2} + 9{y^2}) (2)(2)
Now, again applying the identity a2−b2=(a+b)(a−b){a^2} - {b^2} = (a + b)(a - b) in equation (2), we get
(4x2−9x2)(4x2+9y2)=(2x−3y)(2x+3y)(4x2+9y2)(4{x^2} - 9{x^2})(4{x^2} + 9{y^2}) = (2x - 3y)(2x + 3y)(4{x^2} + 9{y^2}) (3)(3)
Equation (3)(3) cannot be further factorized because this equation has no more linear factors.
Hence the factor of algebraic expression 16x4−81y4=(2x−3y)(2x+3y)(4x2+9y2)16{x^4} - 81{y^4} = (2x - 3y)(2x + 3y)(4{x^2} + 9{y^2}).

Note: For solving this type of question in which we are asked to find the factor of algebraic expression having the knowledge of some basic algebraic identities are must such as a2−b2=(a+b)(a−b){a^2} - {b^2} = (a + b)(a - b),
(a+b)2=a2+2ab+b2{(a + b)^2} = {a^2} + 2ab + {b^2} etc.
To solve these types of questions we just have to break the expression using knowledge of exponents and apply known algebraic identities.