Completely factor nine 𝑚 to the power of four minus 64𝑛 to the power of four.
And what we need to notice here is that nine is three squared, 𝑚 to the power of four is 𝑚 squared squared, 64 is eight squared, and 𝑛 to the four is 𝑛 squared squared. So we can think of this as three times three times 𝑚 squared times 𝑚 squared or, in fact, three 𝑚 squared squared. Likewise, eight squared times 𝑛 squared squared means eight times eight times 𝑛 squared times 𝑛 squared. And we can rearrange that, and it’s 18 squared all squared.
So we can rewrite the whole expression nine 𝑚 to the power of four minus 64𝑛 to the power of four as three 𝑚 squared all squared minus eight 𝑛 squared all squared. And hopefully, you’ll recognize this form as the difference of two squares. It’s something all squared take away something else all squared. Now if you remember, the general format for this in terms of factoring it is 𝑎 squared minus 𝑏 squared, can be factored as 𝑎 minus 𝑏 times 𝑎 plus 𝑏, and that makes sense if you think about it. Let’s multiply it out. 𝑎 times 𝑎 is 𝑎 squared, positive 𝑎 times positive 𝑏 is positive 𝑎𝑏, negative 𝑏 times positive 𝑎 is negative 𝑏𝑎 which we can rearrange as negative 𝑎𝑏, and negative 𝑏 times positive 𝑏 is negative 𝑏 squared.
So we’ve got 𝑎 squared plus 𝑎𝑏 minus 𝑎𝑏. So if we take- start off with 𝑎𝑏 and take away 𝑎𝑏, we’ve got nothing. So we can just cancel those two out like this. And then on the end, we’ve got to take away 𝑏 squared, or negative 𝑏 squared. So this is just 𝑎 squared minus 𝑏 squared. So in our case, 𝑎 squared is three 𝑚 squared squared, so 𝑎 is just three 𝑚 squared. And 𝑏 squared is eight 𝑛 squared all squared, so 𝑏 is just eight 𝑛 squared.
So we can replace 𝑎 and 𝑏, and our factored form becomes three 𝑚 squared minus eight 𝑛 squared times three 𝑚 squared plus eight 𝑛 squared. Although of course, you can write that the other way round. So you can put three 𝑚 squared plus eight 𝑛 squared first and three 𝑚 squared minus eight 𝑛 squared second.
So either one of those two is our completely factored form of nine 𝑚 to the power of four minus 64𝑛 to the power of four.