Question

factorise x⁶y² - y⁶x²​

Answer 1

To factorize : $x^{6}y^{2} - y^{6}x^{2}$

Solution :

• To factorize $x^{6}y^{2} - y^{6}x^{2}$ we use power of product property of exponents .
• The law states to distribute exponent to each multiplicand of product.

        hence write the given expression as:

                          $( x^{3}y) ^{2} - (y^{3}x) ^{2}$

• Now , after applying the law we can see that expression can be further factorized by applying the identity :  

                          $(a^{2}) - (b^{2}) = ( a + b ) (a - b )$

• here , $a = ( x^{3}y)\ and \ b = (y^{3}x )$
• Therefore, applying above identity we get ,

                           $( x^{3}y) ^{2} - (y^{3}x) ^{2}$  $= [(x^{3}y) + (y^{3}x ) ][(x^{3}y - (y^{3})]$

       Hence , factors of $x^{6}y^{2} - y^{6}x^{2}$ are $[(x^{3}y) + (y^{3}x ) ][(x^{3}y - (y^{3})]$ .

Answer 2

Given,

$x^{6}y^{2}-y^{6}x^{2}$

We have to factorize the given expression

Here we have to use power of product property of exponent property

• The law states that the product of powers property states that when multiplying two exponents with the same base, you can add the exponents and keep the base.

We can written as

$(x^{4}y)^{2}-(xy^{4})^{2}$

The above expression same like $a^{2}-b^{2}= (a-b)(a+b)$

Here, $a= x^{4}y$ and $b= y^{4}x$

So, we have to expand the above expression like above identity

$x^{6}y^{2}-y^{6}x^{2}=(x^{4}y-y^{4}x)(x^{4}y+y^{4}x)$

Therefore, the factors of $x^{6}y^{2}-y^{6}x^{2}$ is $(x^{4}y-y^{4}x)(x^{4}y+y^{4}x)$.