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### Course: Algebra basics>Unit 1

Lesson 4: Square roots

# Simplifying square-root expressions: no variables

When square roots have the same value inside the radical, we can combine like terms. First we simplify the radical expressions by removing all factors that are perfect squares from inside the radicals. Then we can see whether we can combine terms or not. If there is only one term, there is nothing to combine.

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• okay so how does 180 times 1/2 = the sqrt of 90 pls help
• Because 180*.5= 90, so finding the square root, it would be the sqrt of 90.
• All you're really looking for are square numbers that can be pulled out of the radical.
An important thing to realize is that sqrt(a•b) = sqrt(a)•sqrt(b). This allows us to separate the radical expression into it's factors. If it has any square factors, they simplify, and you're left with a simplified expression.
Here's an example with actual numbers:
sqrt(12) = sqrt(4•3) = sqrt(4)•sqrt(3) = 2sqrt(3)
• By , you can see the final answer, but I got 6√6 over 9. I got this by simplifying √128, then multiplying the whole fraction by √27 because a radical sign should never be on the denominator. Then after some simplifying, I got 6√6 all over the denominator 9. But I don't understand what I got wrong. Please help!
• I'm going to try and repeat your steps...
1) `√128 = 8 √2` (I think your error could be here)
2) `8 √2 / √27 * (√27/√27)` = `8 √54 / 27`
3) `8 √54 / 27` = `8 √(9*6) / 27` = `8*3 √6 / 27` = `8 √6 / 9`

Either you didn't get the "8" out of √128, or you lost it somewhere along the way.

A couple of tips:
1) Try to reduce the fraction 1st. You can usually save your self quite a bit of work.
2) You did a better job then Sal in trying to get to a complete answer. Sal's answer would typically be considered incomplete as he didn't rationalized the denominator.

Any way, hope this helps.
• I don't understand the fraction part.
• I need help.
• me to
• Why are we learning to simplify square roots that require factoring in algebra basics while factoring isn't covered until Algebra 1?
• Please post questions once and be patient. Questions are answered by other KA users in their spare time. Prime factorization is a pre-algebra topic: https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples

Maybe you need to backup and review basic math and pre-algebra before doing Algebra.
• what if its 4.32 instead of 2.64 in the last question
• The last problem has 2 times 64, not a decimal number of 2.64. And, I'm assuming you means 4*32, not the decimal number 4.32. Use the * (shift 8) for a raised dot to indicate multiplication when typing.

Sal started with 128. He split it into 2(64) because 64 is the largest perfect square. And, all perfect square factors must be simplified out of the square root. If you start with 4(32), you can take the square root of 4, but you can leave the problem as 2 sqrt(32). 32 still contains factors that are perfect squares. You need to find all of them. You could do 2(16) and do the square root of the 16 as well as your original 4. If you don't see the 16, take out another 4, you'll have 128 = 4*4*8. Then split up the 8 to get 128 = 4*4*4*2. Each of those 4's is a perfect square and needs to be simplified by taking their square roots.

Hope this helps.
• 40 is also divisible by 2,but you also said 4 is divisible by 40,dose it matter,is the answer going to be same?
• 40 does not divide evenly into 4. I think you meant to say "40 is divisible by 4".

When simplifying square roots, you need to find perfect square factors and take their square root. You leave any factors inside the radical that are not perfect squares.

Yes, you can start by dividing 40 by 2, but 2 is not a perfect square. So, you need to keep factoring. If needed, you can factor down to prime factors: 40 = 2*2*2*5. Any pair of matching factors creates a perfect square. Over time, you would recognize the 4 is the perfect square factor, and you would factor out the 4 diretly.

Using the prime factors: 2*2*2*5, split them up. (2*2)(2*5). The (2*2) is the perfect square. So, you can take its square root: sqrt(2*2) = 2. And, you would leave (2*5) inside the radical. This makes the final answer:
sqrt(40) = sqrt(2*2) sqrt(2*5) = 2 sqrt(10)

Hope this helps.