Identities with complex numbers
Current time:0:00Total duration:2:32
Voiceover:Let's see if we can factor 36a to the eighth, plus 2b to the sixth power and I encourage you to pause the video and try it out on your own. So let's see if we can express this, or re-express this as the difference of squares using imaginary numbers. So we can rewrite 36 as, six squared, and a to the 8th is the same thing as, a to the fourth squared, and so let me actually just rewrite it this way. We can rewrite it as 6a to the fourth squared. That's this first term right over here. And then the second term we can write it as, actually just let me write it this way first. Let me just write it as a square. Plus, the square root of 2, b to the third power, squared. Now we wanted to write it as a difference of squares. So instead of writing it this way, let's get rid of this plus, and let's - so let me clear that out, and I could write it as subtracting a negative one times that, and negative one, we know, is the same thing as i squared, so we can rewrite this whole thing as 6a to the fourth, squared. And then we have this minus right over here, minus. And so this is i squared. Negative one is i squared. So we can rewrite this in this pink color as i times the square root of two, times b to the third, all of that squared. Notice i squared is negative one. Square root of two squared, is two. B to the third, squared, is b to the sixth power. If I raise something to an exponent, and then raise it to another exponent, I would multiply the two exponents. And so now I've expressed it as a difference of squares, so we're ready to factor. This is going to be equal to 6a to the fourth, minus i times the square root of two, times b to the third, times - and let me get myself space here, times 6a to the fourth, plus all of this business, i times the square root of 2, times b to the third power, and we are done.