How to Simplify Square Root

Understand what factor analysis is. The goal of square root reduction is to rewrite it in a simpler and easier to understand form to solve math questions. Factorization is a way of dividing a larger number into many smaller factors , for example splitting 9 into 3 x 3. Once we have found the factors of the number in question, we can rewrite the square root of the number. that into a simpler form, maybe even an integer. For example √9 = √(3×3) = 3. The steps below will show you the process of reducing more complex square roots.

Divide the number under the root by the smallest possible prime. If the base part is even, divide by two. If it’s an odd number, try to see if it’s divisible by 3. In case the number below the root is not divisible by both 2 and 3, proceed to divide by the next prime numbers in the list below until you find the smallest prime divisor of the number under the root. We only consider prime numbers because all other numbers can be factored as the product of a prime number. For example, we’re not going to divide the bottom by 4, because any number that’s divisible by 4 is also divisible by 2.

• 2
• 3
• 5
• 7
• 11
• 13
• 17

Rewrite the square root as a multiplication problem. Keep all the factors under the root sign. For example, when we reduce √98, we find that 98 ÷ 2 = 49, so 98 = 2 x 49. So we can rewrite it as: √98 = √(2 x 49).

Repeat the above steps with the remaining multipliers. Before reducing the square root in question, we need to multiply the factors until we get the result of the analysis as two identical numbers. Recalling the meaning of the square root, this makes perfect sense: since √(2 x 2) means “the number that, when multiplied by itself, is 2 x 2.” And obviously in this case it’s the number 2. Similarly, we repeat these steps with our example √(2 x 49):

• We have factored 2. (In other words, this is one of the prime numbers listed above). Therefore, we will ignore this number and continue to split 49 into smaller factors.
• 49 is not divisible by 2, 3, or 5. We can check by using a calculator or doing division. Since the result of dividing 49 by 2, 3 or 5 does not give us an integer, we will ignore these numbers and continue to divide.
• 49 is divisible by 7. We have 49 ÷ 7 = 7, that is 49 = 7 x 7.
• Rewrite the problem, we get: √(2 x 49) = √(2 x 7 x 7).

“Draw” a number from the radical sign. Once we have factored the number in question, including two identical numbers, we can pull the number out of the radical sign. All other factors remain under the root sign. For example: √(2 x 7 x 7) = √(2)√(7 x 7) = √(2) x 7 = 7√(2).

• We can stop the analysis when we have found two identical factors. For example √(16) = √(4 x 4) = 4. If we continue the analysis, the final result remains the same, except that we have to perform the division more times: √(16) = √(4 x 4) = √(2 x 2 x 2 x 2) = √(2 x 2)√(2 x 2) = 2 x 2 = 4.

If the number of factors under the root is more than one, multiply them together. With large square roots, we can perform the reduction many times. In that case, multiplying the factors will give the final result. Consider the example below:

• 180 = (2 x 90)
• 180 = (2 x 2 x 45)
• √180 = 2√45, but the remaining root can still be further factored into smaller factors
• 180 = 2√(3 x 15)
• 180 = 2√(3 x 3 x 5)
• √180 = (2)(3√5)
• √180 = 6√5

Write “nonreducible” if the factorization does not give two numbers the same. Some square roots are already in their reduced form. If we continue the analysis until all the factors below the root are prime (mentioned in the previous steps) and no two numbers are the same, then we cannot simplify any further. Maybe the topic under consideration is just a tip! For example, let’s 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 numbers are prime, so we can’t simplify any further. In addition, these three numbers are all different, so we can’t pull any of the three numbers out of the radical sign. So √70 cannot be reduced any more.