# Question Video: Factoring a Difference of Sixth Powers Mathematics

Factor the expression 𝑥⁶ − 𝑦⁶.

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### Video Transcript

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 squared.

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.