Week 4 – Simplifying Radical Expressions

Posted on by shannonf2015

I learned how to expand and simplify a radical expression.

(3\sqrt{5} + 7\sqrt{8})(\sqrt{10} – 6\sqrt{5})

There are multiply ways of starting to simplify this expression, but the first thing that I do is see if I can simplify the radicals given.

Looking at the expression, I can see that I can simplify 7\sqrt{8}, as 8 has a square root, which is 4.

7\sqrt{8} —>  7\sqrt{{4}\times{2}} —> 7(2)\sqrt{2} —> 14\sqrt{2}

The next step is to distribute the values in the first bracket and multiply with all the values in the second bracket.

This process is called FOILING.

F – first terms

O – outer terms

I – Inner terms

L – left over terms

Rule: We can simply multiply the radicals together, unlike adding and subtracting where they have to have the same index and radicand.

So in my example, it doesn’t simplify by a lot as they don’t have any like terms.

The answer would therefore be :  -90 + 15\sqrt{2} + 28\sqrt{5} – 84\sqrt{10}

Since my example isn’t the best to show how to simplify when it comes to like terms, let’s look at this example.

I did this equation in two ways which does give you the same answer. The first way, I simplified the radicals first, then added the like terms together. Note: They have to have the same radicand and same index in order to add them or subtract them together.

The second method, I just added the like terms together first as 4\sqrt{x^3} had the same radicand and index as -7\sqrt{x^3} and when you simplify 2\sqrt{x^2}, it becomes 2x which can be added with 3x. So when you add the coefficients together keeping the same radicand and index, you are combining like terms.

Then, I simplified the radicals, {-3}\sqrt{x^3} can be simplified to {-3x}\sqrt{x}.

Since we have a variable in our expression, we must define x. Our index here is 2 (square root), therefore, x must be greater than or equal to 0 because we cannot have two numbers that are the same that will multiply to give a negative number: x ≥ 0

Last thing that I have learned was simplifying fractions with a radical as the denominator.

Since we cannot leave radicals in the denominator, we must rationalize the denominator in order to make it a real number.

Rationalizing the denominator means multiplying both the nominator and denominator by the denominator. I included the coefficient but you don’t have to. It would probably be best to just multiply \sqrt{5} with the denominator and nominator, but both give the same results.

After I multiplied them, I noticed I could simplify even more. Then finally, I simplified the coefficients as both can be divide by 10.