Video Transcript
Simplify 𝑥 to the sixth power over 𝑥 to the sixth power.
Well, when we’re looking at 𝑥 to the sixth power over 𝑥 to the sixth power, it can also be said as 𝑥 to the sixth power divided by 𝑥 to the sixth power. There’re a couple of ways that we could solve this problem. Well, the first method is by using a couple of our exponent rules. So let’s have a look at which rules we can use. Well, the first exponent rule we’d use is one where we’ve got 𝑥 raised to the power of 𝑎 divided by or over 𝑥 raised to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏.
And what I can quickly do is demonstrate why this would work. So if we had 𝑥 raised to the power of four over 𝑥 raised to the power of three. Well, this would be the same as 𝑥 multiplied by 𝑥 multiplied by 𝑥 multiplied by 𝑥 over 𝑥 multiplied by 𝑥 multiplied by 𝑥. Well, we could divide through then both the numerator and denominator by 𝑥 and 𝑥 again and 𝑥 once more. And what we’d be left with is just 𝑥. Well, 𝑥 itself is the same as 𝑥 raised to the power of one. And if we do four minus three, we get one. Okay, great. So that’s the first exponent rule we’re gonna use. Now, let’s use it with our example.
Well, if we use our rule, we’re gonna subtract our exponents. So we’re gonna have 𝑥 raised to the power of six minus six, which gives us 𝑥 to the power of zero. But what is the value of 𝑥 to the power of zero? Well, here we can use another rule that we’ve got for exponents which tells us that if we have anything to the power of zero, it’s equal to one. I’ve written here 𝑎 to the power of zero. But it could be any letter. I’ve just chosen that so that it’s different to the 𝑥 that we’ve got in our question.
And to give an example of why it works, if you take a look at three. If we got three cubed, it’s equal to 27. Three squared is equal to nine. Three to the power of one or just three is equal to three. Well, every time we reduce the exponent by one, what we do is divide the answer by three. So we have 27 divided by three is nine. Nine divided by three is three. So therefore, if we go from three to the power of one to three to the power of zero. Then what we’re gonna do is divide the result of three to the power of one, which is three, by three, which gives us one. So therefore, we can say that if we simplify 𝑥 raised to the power of six over 𝑥 raised to the power of six, the answer is gonna be one.
Now, I did say that there is another way to solve it that didn’t involve our exponent rules. What if we inspect our expression that we’re trying to simplify? We could see that the numerator and denominator are the same. So therefore, if you divide something by itself, then the result is always going to be one. So we could’ve done it that way. So therefore, we definitely know that if we simplified 𝑥 raised to the power of six over 𝑥 raised to the power of six, the answer is going to be one.
