Video Transcript
Simplify two root five to the power of negative two multiplied by root two to the power of negative eight over root five to the power of negative two.
Well, if we’re going to simplify our expression, the first thing we can do is deal with this first set of parentheses here. And what we can do is split it up. So we can say that two root five to the power of negative two is the same as two to the power of negative two multiplied by root five to the power of negative two. So now, if we substitute this back into our original expression, what we’re gonna get is two to the power of negative two multiplied by root five to the power of negative two multiplied by root two to the power of negative eight all over root five to the power of negative two.
So why have we done this? Well, we’ve done this cause we can see clearly now that we can cancel because what we can do is divide both the top and bottom, so the numerator and the denominator, by root five to the power of negative two. And if we do this, what we’re left with is two to the power of negative two multiplied by one multiplied by root two to the power of negative eight over one, which we can simplify down to two to the power of negative two multiplied by root two to the power of negative eight. So what we can do now to work this out is utilize a couple of our exponent rules. The first one is that if we have root 𝑥, this is equal to 𝑥 to the power of a half. So if we have the root of any value, then this is equal to that value raised to the power of a half.
So then we can move on to another exponent rule. And that tells us that if we got 𝑥 to the power of 𝑎 to the power of 𝑏, this is equal to 𝑥 to the power of 𝑎𝑏. So we multiply our powers or exponents. So then in our example, what we’re gonna do is multiply a half by negative eight. So we’re gonna now have two to the power of negative two multiplied by two to the power of negative four. So now, we can utilize another one of our exponent rules. And that one is that if we have 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏, then this is equal to 𝑥 to the power of 𝑎 plus 𝑏. So we add our exponents. So this is if we’ve got the same base and we multiply it. Then what we do is we add the exponents.
So in our case, what we’re gonna do is add negative two and negative four. Well, negative two add negative four is negative six. So we’ve now got two to the power of negative six. Have we finished here? Well, no. We can actually use one more of our exponent rules. And that rule is that 𝑥 to the power of negative 𝑎 is equal to one over 𝑥 to the power of 𝑎. So therefore, we can say that two to the power of negative six is the same as one over two to the power of six. Well, we can work out what two to the power of six is. Well, two to the power of six is 64. And that’s cause two multiplied by two is four, then another two is eight, multiplied by another two, 16, another two, 32, and, finally, another two, 64. So therefore, we could say our final answer is one over 64.
