Week 2 – Precalculus – Converting Entire Radicals to Mixed Radicals

Hello! This week we learned how to convert entire radicals ( ex. ) to mixed radicals ( ex. ). I wanted to cover this topic specifically because it was something that I struggled with initially and it will be good to have something that I can look back on if I do get stumped on a question concerning the conversion between entire radicals and mixed radicals in the future.

Say you wanted to convert to a mixed radical.

The key is looking for a perfect square factor that the number has. For this number, the perfect square factor would be 4. 4 multiplied by 3 yields 12, so our pair would be 4 and 3.

I find it easier to write these two numbers as two separate square roots: multiplied by . We know the square root of 4, as it is a perfect square, so we simplify that down to 2 . We cannot evaluate to a whole number, so we will leave it as it is.

Place those two next to each other, and you’ve got your first mixed radical! For reference, the finished product should look like this: .

Now on to a harder question.

Say you wanted to convert to a mixed radical.

As stated previously, look for a perfect square factor that the number has. For this particular question, there are two perfect square factors that has: 24 and 4, and 16 and 6. You want to take the largest perfect square when converting an entire radical to a mixed radical because it breaks up what is going to be left underneath your radicand better.

(P.S If you are having trouble finding a perfect square factor, don’t be afraid to make a factor tree!)

So as we have done before, write this pair as two separate square roots: and . We know that the square root of 16 is 4, so we will simplify the square root of 16 to 4. We cannot evaluate to a whole number, so we will leave it as it is.

Place those two next to each other, and you have your second mixed radical! For reference, the finished product should look like this: .

Now, cube roots are a little different.

Instead of looking for a perfect square factor, you will be looking for a perfect cube factor. 

Say you wanted to convert to a mixed radical.

First, you would ask yourself: what is the largest perfect cube factor that factors in to ? Then you would create a factor tree or do what helps you find that factor, and you would eventually find the pair of 8 and 6. 8 multiplied by 6 yields 48, so that is the pair that we will be using.

Write this pair as two separate cube roots: and . We know that the cube root of 8 is 2 because it is a perfect cube, so we will simplify the cube root of 8 to 2. We cannot evaluate to a whole number, so we will leave it as it is.

Place those two next to each other and you have a cube root mixed radical! For reference, the finished product should look like this: .

Hope this helped!