How To Add Fractions With Radicals In The Denominator

- we want to rationalize the denominators of the given radical expressions. to do this, we would eliminate these square roots from the denominator. but let's first start by simplifying the denominators.

How To Add Fractions With Radicals In The Denominator, looking at our first fraction, the denominator will simplify because it does contain a perfect square factor. this is equal to 5 over the square root of 3 and then because we have the square root of x cubed,

we can write x cubed as x to the second x x. so because x squared is a perfect square, this will simplify. so now we have 5 square root 3 divided by x square root x. but we still have to rationalize the denominator. we don't want to have the square root x here. so notice how if we had one more factor of x underneath the square root it would simplify perfectly

and eliminate the square root from the denominator. so what we'll do here is multiply both the numerator and denominator by the square root of x. notice how this is like multiplying by 1 but now we're going to have the square root of x squared when we find this product. so now we'll go ahead and multiply, the numerator is going to be 5 x the square root of 3 x x or 3x, and now the denominator is going to be x

x the square root of x x x. again this is a perfect square factor and therefore simplifies. so finally we have 5 square root 3x divided by-- well this simplifies to x but we already have a factor of x here, so we'd have x squared. now remember these x's here do not simplify because this x is underneath the square root and these aren't. let's take a look at another example.

let's start by simplifying the square root of 48. to simplify the square root of 48, we could write out the prime factorization of 48 to identify the perfect score factors but if we recognize that 48 is equal to 3 x 16, that would save us some time. we'd have 6 all over the square root of 3 x 16 but 16 is a perfect square so this would simplify to 6 all over 4 square root 3.

now we could simplify the 6 and the 4 let's rationalize the denominator now. the denominator would simplify perfectly if we had the square root of 2 factors of 3 and since we only have 1 factor of 3 we're going to multiply both the numerator and denominator by the square root of 3 which is like multiplying by 1 but now when we multiply we have 6 square root 3, all over 4

x the square root of 3 x 3 and now we have a perfect square here. so this will simplify and eliminate the square root. so this would be 6 square root 3 all over this would be 4 x 3 which of course is 12 so we have 6 square root 3 over 12 but now notice how we have a common factor of 6 here. there's one factor of 6 and 6 and 2 factors of 6 and 12 so this finally simplifies to the square root of 3 over 2.

i do realize you may be able to skip some of these steps but think when you're first learning how to do this, it is nice to see a little extra work. next we'll talk about how to rationalize a denominator when we have a sum or difference involving a radical in the denominator. i hope you found this helpful. â