What's a "Simplified" Radical Expression?
There are certain rules that you follow when you simplify expressions in math. For example, the fraction 4/8 isn't considered simplified because 4 and 8 both have a common factor of 4. You also wouldn't ever write a fraction as 0.5/6 because one of the rules about simplified fractions is that you can't have a decimal in the numerator or denominator.
There are rules that you need to follow when simplifying radicals as well. One rule is that you can't leave a square root in the denominator of a fraction. Another rule is that you can't leave a number under a square root if it has a factor that's a perfect square. If a number inside a square root has a factor of 4, 9, 16, 25, 36, 49, etc., you'll have to do some steps to simplify the radical. We'll show you how to do this next.
How To Simplify Radicals
Here's an important property of radicals that you'll need to use to simplify them. It says that the square root of a product is the same as the product of the square roots of each factor.
When you write a radical, you want to make sure that the number under the square root sign doesn't have any factors that are perfect squares. Open the next tab to see a few examples.
There are a couple different ways to simplify this radical. You could start by doing a factor tree and find all the prime factors. Or you could start looking at perfect square and see if you recognize any of them as factors. The smallest perfect square is 1, but it doesn't do much good to factor out a 1. So start with the next perfect square: 4. Can you divide 50 by 4? No. Try the next few: 9, 16, 25, 36, 49 . . . . Which one is a factor of 50? The 25 is. We can rewrite 50 as the product of 25 and 2. Then we'll be able to use the product property above to simplify the square root.
The square root of 25 is just 5, so we can simplify the square root in the front. The second square root just has a 2 inside. 2 doesn't have any factors that are perfect squares other than 1, so that part we just leave as it is since it can't be simplified any more.
Think about the factors of 63. Start with the smallest perfect square and work your way up. Can you divide 63 by 4, 9, 16, 25, 36, 49, etc.? You can divide 63 by 9 and 9 is a perfect square. We can rewrite 63 as the product of 9 and 7 and split this problem up into two radicals.
7 doesn't have any factors that are perfect squares other than 1, so it's left under the radical sign.
You can also simplify radicals with variables under the square root. You'll want to split up the number part of the radicand just like you did before, but you'll also split up the variables too. With variables, you can only take the square root if there are an even number of them. If there's a variable to an odd exponent, you'll have a variable left over inside the radical.
Start with the 24. Do you know any factors of 24 that are perfect squares? 4 is a factor, so we can split up the 24 as a 4 and a 6. The 6 doesn't have any factors that are perfect squares so the 6 will be left under the radical in the answer.
Next look at the variable part. We can only take the square root of variables with an EVEN power (the square root of x squared, x to the 4th, x to the 6th, etc.) But we can split up the x cubed as x squared times x. Then we'll be able to take the square root of x squared and the single x leftover will go with the 6 inside the square root in the answer.
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