Can you calculate the root of any perfect-square factors? Can you rewrite the expression so that it does not have a radical in the denominator?
See solution.
Practice makes perfect
A radical expression is an expression that contains a radical. Let's look at a few examples.
2 sqrt(3), 5/sqrt(4), 8 sqrt(13)/7
Are all of these expressions written in simplified form? To answer that question, let's use the following information. A radical expression is simplified if all of the following statements are true.
No radicals appear in the denominator of a fraction.
Let's see if each of these statements is true for our examples.
Example
2sqrt(3)
5/sqrt(4)
8sqrt(13)/7
1. Does the radical have any perfect-square factors other than 1?
No
Yes, it has another perfect-square factor of 4=2^2.
No
2. Does the radical contain any fractions?
No
No
No
3. Do any radicals appear in the denominator of a fraction?
No
Yes, sqrt(4).
No
As we can see, the first and the third radical expressions, 2sqrt(3) and 8sqrt(13)7, meet all of the requirements. Therefore, they are in simplified form. However, the second one does not. Let's simplify it by rationalizing the denominator. We will multiply the fraction by sqrt(4)sqrt(4).
5/sqrt(4)* sqrt(4)/sqrt(4)=5sqrt(4)/4
Now, the third requirement is met for this expression too. We can also calculate the square root of 4. This way we will get rid of the perfect-square factors in the radical.
5sqrt(4)/4=5* 2/4= 2.5
Finally, all of our radical expressions are in simplified form.
