This week in Math 11 we have continued adding things to radicals step by step. However, something I’ve learned that I could reflect on is how to solve numbers with fractions as exponents. We have reviewed our exponent laws from last year to refresh our minds because it will serve us well in future lessons. I do recall exponent laws from last year, but the new thing I had never seen until this week was numbers with fractions exponents that were related to radicals. Let me share the knowledge I’ve gained this week it’s not a lot, but will be good to know for the future.

Before we begin let us refresh our minds to remember these important Exponent Laws: 

Now, that we have recalled our lesson from the previous year on exponents.

Let’s Start with a Simple Examples:

In this example we have a 4^1/2 we need to evaluate it. The 4 becomes our radicand and the 2 in the denominator will be our root. In this example, the square root of four is √4=2. For the 1 in the numerator, we don’t have to worry about it right now, but later if it’s not one you have to put brackets around the square root and the radicand inside and put whatever number you have in the numerator as a power to multiply everything at the end to find the answer. We’ll see that later on with more examples.

This is another example related to the first example just with different numbers here we have 100^1/2 we turn our 100 into a radicand and our denominator will be the root of the 100 which is a square root so the √100=10.

Here is another example just with a different denominator. Here we have 125^1/3. We take 125 and turn it into our radicand, the denominator in the fraction we started with is 3 which will be our root. So now we have the cube root of 125=5.

Extra Work with a Numerator and a Denominator

Here we move into a more complicated example where we have a higher numerator. We start with 9^3/2 just like we did before turning 9 into a radicand the 2 in the denominator into a root and the new step will be dealing with the numerator when it’s not 1. Here we add brackets around the whole equation we have and the 3 in the numerator will be the power of the whole thing just like it’s coded in red in the photo. Until now we have gotten to (√9)^3 now you can find the square root of 9 and the 3 stays as the power we just move the brackets because there is nothing except 9, but if there is a negative or a coefficient inside the bracket before moving it then you have to keep the brackets because the power is applied to everything inside the brackets.

Dealing with Negative Fractions:

Now that we have gotten the idea of evaluating simple examples how about we try examples with negative fractions? Here we have a 9^-1/2 but the negative sign is in the middle along with the division line as shown in the photo. What we do here is move the negative sign to the numerator just like we did with fractions back in grade 10. Then we turn our denominator into our root which will become √9. But we also have a negative 1 in this example the 1 in the numerator is important in this case because we have a negative there so we follow the same steps we did before adding brackets and the -1 becomes the power that goes outside the brackets. After this, we end with (√9)^-1 solve what’s in the brackets first √9=3. Then don’t forget about the -1 hanging on the outside move it with the 3 so we have 3^-1 this will equal -3, but remember that we can do reciprocal so -3 is the same thing as -3/1 if we flip the number we would end up with 1/3 as our finale answer which is correct.

Now try 64^-5/6 on your own the answer key is at the very bottom:

A HINT I’ve got from Ms. Burton and want to share with you because it might help you as well:

 

The meaning of this hint is that the roots are always at the bottom in previous examples you might be still confused if the denominator or numinator is the square root. Well, this hit tells you the root is always at the bottom. If you plug that into one of the questions let’s recall the first example 4^1/2 in the exponent fracion 2 is our root and that’s because it’s on the bottom. Hope that helps solve some struggles.

 

Answer Key: