Factor the expression 𝑥 raised to
the sixth power minus 𝑦 raised to the sixth power.
In this question, we are asked to
factor an algebraic expression. We can first note that we have the
difference between two exponential expressions. There are two ways that we can
factor this expression. We could note that we have a
difference between two squares or that we have a difference between two cubes. Either method will work. We will rewrite the expression as
the difference between two squares, that is, 𝑥 cubed squared minus 𝑦 cubed
We can then factor this expression
by recalling the formula for factoring the difference between squares. We have that 𝑎 squared minus 𝑏
squared is equal to 𝑎 minus 𝑏 times 𝑎 plus 𝑏. In our case, the value of 𝑎 is 𝑥
cubed and the value of 𝑏 is 𝑦 cubed. Substituting 𝑎 equals 𝑥 cubed and
𝑏 equals 𝑦 cubed into the difference between squares formula gives us 𝑥 cubed
minus 𝑦 cubed times 𝑥 cubed plus 𝑦 cubed.
We should attempt to factor the
expression fully. So we need to see if we can factor
further. We can see that we are multiplying
a difference of cubes by a sum of cubes. So we can factor further by
recalling how to factor these types of binomials. First, we recall that 𝑎 cubed
minus 𝑏 cubed is equal to 𝑎 minus 𝑏 times 𝑎 squared plus 𝑎𝑏 plus 𝑏
squared. We can use this to factor 𝑥 cubed
minus 𝑦 cubed. We set 𝑎 equal to 𝑥 and 𝑏 equal
to 𝑦 to obtain 𝑥 minus 𝑦 times 𝑥 squared plus 𝑥𝑦 plus 𝑦 squared. We can factor the sum of two cubes
by recalling that 𝑎 cubed plus 𝑏 cubed is equal to 𝑎 plus 𝑏 times 𝑎 squared
minus 𝑎𝑏 plus 𝑏 squared. This allows us to factor 𝑥 cubed
plus 𝑦 cubed. We set 𝑎 equal to 𝑥 and 𝑏 equal
to 𝑦 to obtain 𝑥 plus 𝑦 times 𝑥 squared minus 𝑥𝑦 plus 𝑦 squared.
We cannot factor any further. However, we can reorder the factors
into any order. Hence, we have shown that the
expression 𝑥 raised to the sixth power minus 𝑦 raised to the sixth power is equal
to 𝑥 plus 𝑦 times 𝑥 squared minus 𝑥𝑦 plus 𝑦 squared times 𝑥 minus 𝑦 times 𝑥
squared plus 𝑥𝑦 plus 𝑦 squared.