Content

• Steps
• Method 1 of 3: Factoring
• Method 2 of 3: Full Square
• Method 3 of 3: Terminology
• Tips
• Warnings

Simplifying the square root is not at all as difficult as it might seem. You just need to factor the number and extract complete squares from the root sign. By memorizing a few of the most common squares and learning how to factor a number, you can easily simplify square roots.

Steps

Method 1 of 3: Factoring

1. 1 The goal of square root simplification is to rewrite it in a form that is easier to use in calculations. Factoring a number is finding two or more numbers that, when multiplied, will give the original number, for example, 3 x 3 = 9. Having found the factors, you can simplify the square root or get rid of it altogether. For example, √9 = √ (3x3) = 3.
2. 2 If the radical number is even, divide it by 2. If the root number is odd, try dividing it by 3 (if the number is not divisible by 3, divide it by 5, 7, and so on along the list of primes). Divide the radical number exclusively by prime numbers, since any number can be decomposed into prime factors. For example, you do not need to divide the radical number by 4, since 4 is divisible by 2, and you have already divided the radical number by 2.
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3. 3 Rewrite the problem as the root of the product of two numbers. For example, simplify √98: 98 ÷ 2 = 49, so 98 = 2 x 49. Rewrite the problem like this: √98 = √ (2 x 49).
4. 4 Continue expanding the numbers until the product of two identical numbers and other numbers remains under the root. This makes sense when you think about the meaning of the square root: √ (2 x 2) is equal to the number, which, if multiplied by itself, will be equal to 2 x 2. Obviously, this number is 2! Repeat the above steps for our example: √ (2 x 49).
   • 2 has already been simplified as much as possible, since it is a prime number (see the list of primes above). So factor 49.
   • 49 is not divisible by 2, 3, 5. So move on to the next prime number - 7.
   • 49 ÷ 7 = 7, so 49 = 7 x 7.
   • Rewrite the problem like this: √ (2 x 49) = √ (2 x 7 x 7).
5. 5 Simplify the square root. Since under the root is the product of 2 and two identical numbers (7), you can move such a number outside the root sign. In our example: √ (2 x 7 x 7) = √ (2) √ (7 x 7) = √ (2) x 7 = 7√ (2).
   • Once you get two of the same numbers under the root, you can stop factoring the numbers (if you can still factor them). For example, √ (16) = √ (4 x 4) = 4. If you continue factoring the numbers, you get the same answer, but do more calculations: √ (16) = √ (4 x 4) = √ (2 x 2 x 2 x 2) = √ (2 x 2) √ (2 x 2) = 2 x 2 = 4.
6. 6 Some roots can be simplified many times over. In this case, the numbers removed from the root sign and the numbers in front of the root are multiplied. For example:
   • √180 = √ (2 x 90)
   • √180 = √ (2 x 2 x 45)
   • √180 = 2√45, but 45 can be factorized and simplified the root again.
   • √180 = 2√ (3 x 15)
   • √180 = 2√ (3 x 3 x 5)
   • √180 = (2)(3√5)
   • √180 = 6√5
7. 7 If you cannot get two identical numbers under the root sign, then such a root cannot be simplified. If you have decomposed the radical expression into the product of prime factors and there are no two identical numbers among them, then such a root cannot be simplified. For example, let's try to simplify √70:
   • 70 = 35 x 2, so √70 = √ (35 x 2)
   • 35 = 7 x 5, so √ (35 x 2) = √ (7 x 5 x 2)
   • All three factors are simple, so they can no longer be factorized. All three factors are different, so you can't move an integer out of the root sign. Therefore, √70 cannot be simplified.

Method 2 of 3: Full Square

1. 1 Memorize a few squares of prime numbers. The square of a number is obtained by raising it to the second power, that is, multiplying it by itself. For example, 25 is a perfect square because 5 x 5 (5) = 25.By memorizing at least a dozen complete squares, you can quickly simplify the roots. Here are the first ten complete squares:
   • 1 = 1
   • 2 = 4
   • 3 = 9
   • 4 = 16
   • 5 = 25
   • 6 = 36
   • 7 = 49
   • 8 = 64
   • 9 = 81
   • 10 = 100
2. 2 If you see a complete square under the square root sign, then get rid of the root sign (√) and write down the square root of that complete square. For example, if the number 25 is under the square root sign, then such a root is 5, since 25 is a perfect square.
   • √1 = 1
   • √4 = 2
   • √9 = 3
   • √16 = 4
   • √25 = 5
   • √36 = 6
   • √49 = 7
   • √64 = 8
   • √81 = 9
   • √100 = 10
3. 3 Decompose the number under the root sign by the product of a perfect square and another number. If you notice that the radical expression can be decomposed into the product of a full square and a number, then you will save time and effort. Here are some examples:
   • √50 = √ (25 x 2) = 5√2. If the radical number ends in 25, 50, or 75, you can always expand it into the product of 25 and some number.
   • √1700 = √ (100 x 17) = 10√17. If the radical number ends in 00, you can always expand it into the product of 100 and some number.
   • √72 = √ (9 x 8) = 3√8. If the sum of the digits of the radical number is 9, you can always decompose it into the product of 9 and some number.
   • √12 = √ (4 x 3) = 2√3. Always check if the radicals are divisible by 4.
4. 4 Decompose the radical number by the product of several complete squares. In this case, take them out from under the root sign and multiply. For example:
   • √72 = √ (9 x 8)
   • √72 = √ (9 x 4 x 2)
   • √72 = √ (9) x √ (4) x √ (2)
   • √72 = 3 x 2 x √2
   • √72 = 6√2

Method 3 of 3: Terminology

1. 1 √ is the square root sign. For example, in √25, “√” is the square root sign.
2. 2 A radical expression is written under the root sign. For example, "25" is a radical expression (number) in √25.
3. 3 The coefficient is the number in front of the root sign (to the left of it). This is the number by which the square root is multiplied; it is written to the left of the √ sign. For example, "7" is a factor of 7√2.
4. 4 A multiplier is an integer that is obtained by dividing another number. 2 is a factor of 8, since 8 ÷ 4 = 2, and 3 is not a factor of 8, since 8 is not divisible by 3 (entirely). 5 is a factor of 25, since 5 x 5 = 25.
5. 5 Understand the meaning of square root simplification. Square root simplification is finding complete squares among the factors of the radical expression and extracting them from under the root. If the number is a perfect square, then the root sign will disappear as soon as you write down its root. For example, √98 can be simplified to 7√2.

Tips

• To find a complete square (as one of the factors of the radical expression), simply look through the list of complete squares, starting with the complete square closest to the radical number (and then in decreasing order). When looking for a complete square in the number 27, start with a complete square of 25, then 16, and stop at 9.

Warnings

• Under no circumstances should you have a decimal!
• Calculators can be useful for calculations with large radical numbers, but it is better to practice simplifying the roots manually.