Question Video: Simplifying Rational Expressions with Fractional Exponents Mathematics

Video Transcript

Express the square root of nine 𝑥 squared over 100 𝑦 to the power of six 𝑧 to the power of four in its simplest form.

In order to solve this problem, we’re gonna have a look at a couple of surd rules. Our first rule that we’re going to use is that root 𝑎 over 𝑏 is equal to root 𝑎 over root 𝑏. I’m gonna use this to express the square root in our question in a different form. You can now see that we’re actually express our square root as root nine 𝑥 squared over root 100𝑦 to the power of six 𝑧 to the power of four. We’re gonna have a look at it in two parts.

I’m gonna start with root nine 𝑥 squared. To simplify root nine 𝑥 squared, we can actually use this surd rule, which is that root 𝑎 multiplied by root 𝑏 is equal to root 𝑎𝑏. So therefore, we can express root nine 𝑥 squared as root nine multiplied by root 𝑥 squared. Now, that suggests that you need to use this step every time, but I just wanted to break it down to steps to see how you get to the final answer. This gives us the answer of three 𝑥 cause we get the three because the square root of nine is equal to three. And we get the 𝑥 because the square root of 𝑥 squared is equal to 𝑥.

And strictly, the square root means only the positive root. So to ensure that this is the case, you put these vertical lines either side of our answer. And what these lines mean is the modulus or absolute value, which will give us the positive value only.

We can now have a look at the denominator which is root 100𝑦 to the power of six 𝑧 to the power of four. Again, to enable us to show how we simplify this, I’ve actually split it down to root 100 multiplied by root 𝑦 to the power of six 𝑧 to the power of four. Again, I use root 𝑎 multiplied by root 𝑏 equals root 𝑎𝑏 to enable me to do this.

In order to simplify it further, we’re going to introduce a couple of exponent rules. The first of which being the 𝑚th root of 𝑎 is equal to 𝑎 to the power of one over 𝑛, which we can use to give us 10 multiplied by 𝑦 to the power of six 𝑧 to the power of four on parentheses to the power of a half. We can then use the second exponent rule, which is 𝑎 to the power of 𝑚 in parentheses to the power of 𝑛 is equal to 𝑎 to the power of 𝑚𝑛, which means that we actually multiply the powers together.

So this is gonna allow us to simplify even further. So we’re gonna get 10𝑦. And then, it is the 𝑦 to the power of six multiplied by a half, which gives us 𝑦 cubed or 𝑦 to the power of three. And then it’s gonna be 𝑧 to the power of four multiplied by a half, which give us 𝑧 squared. And again, we include our vertical lines either side as we want the absolute value or modulus.

So now, we can move back over to the left-hand side, where we’re gonna get a final answer, where we can say that if we express the square root of nine 𝑥 squared over 100 𝑦 to the power of six 𝑧 to the power of four in its simplest form, we’re gonna get the modulus of three 𝑥 over 10 𝑦 cubed 𝑧 squared.