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In mathematics, squares and square roots are inverse operations. The Square of a number is actually the value of power 2 of the number. Whereas, the square root of a number is the number that we require to multiply by itself to obtain the original number. For simplifying square roots with variables, if m is the square root of n, it implies that m × m = n. The square of any number is always a positive number, thus every number contains two square roots, one positive value, and one negative value.

As already discussed, the square root is an inverse operation of squaring a number. The square root of a number m is represented using a radical sign √x or x ½. A square root of a number m is such that, a number n is the square of m, simply written as n² = m.

For example, the square root of 49 is denoted as √49 = 7. A number whose square root is computed is called a radicand. In this expression, √49 = 7, number 49 is the radicand.

There are different techniques of simplifying square roots with variables or complex expressions, based upon the number of radicals and the values under each radical.

Simplification of the square root of a number entails different methods that are as follows:

Prime Factorization Method

Square Root by Prime factorization of any number means to denote the number as a product of prime numbers. To simplify the square root of a given number by prime factorization method, you need to follow the steps below:

Step 1: Divide the given number into its prime factors.

Step 2: Form pairs of similar factors in a way both factors are equal to each pair.

Step 3: Take out one factor from the pair.

Step 4: Determine the product of the factors obtained by taking one factor from each pair.

Step 5: That product is the square root of the given number.

Let’s find the square root of 144 by prime factorization method:

144 = 2 × 2 × 2 × 2 × 3 × 3

Now pairing the factors;

(2 × 2) × (2 × 2) × (3 × 3)

= 2² × 2² × 3²

= (2 × 2 × 3)²

= 12²

144 = 12²

√144 = 12

Repeated Subtraction Method

The Repeated Subtraction Method of Square Root is very simple. You will require subtracting the consecutive odd numbers from the number for which we are calculating the square root, till you reach 0. The number of times you subtract is the square root of the given number; however, it works only for perfect square numbers. Let us see how the square root of 16 is simplified using this method:

16 – 1 = 15

15 – 3 = 12

12 – 5 = 7

7 – 7 = 0

You can see that we have subtracted 4 times. Therefore, √16 = 4.

Estimation Method

Square Root by estimation indicates a reasonable approximation of the actual value in order to make calculations simpler and realistic. This technique helps in guessing the square root of a given number. Let’s apply this method to find √15. To find the nearest perfect square number to 15, we are familiar with √16 = 4 and √9 = 3. This states that √15 lies between 3 and 4. Now, we need to check if √15 is closer to 3 or 4. Let us consider 3.5 and 4. Thus, √15 lies between 3.5 and 4 and is more close to 4.

Now, consider squares of 3.8 and 3.9. 3.82 = 14.44 and 3.92 = 15.21. This states that √15 lies between 3.8 and 3.9. repeating the process, we see 3.85 is the nearest ant thus √15 = 3.872.

Long Division Method

Square Root by Long Division Method is a technique for dividing large numbers into parts, splitting up the division problem into an order of easier steps. We can find the exact square root of any given number using this method. Let's find and simplify the square root of 180 using the long division method.

Step 1: Place a bar over every pair of the number beginning from the unit’s place (right-most side). You will have two pairs, i.e. 1 and 80.

Step 2: We divide the left-most number by the largest number whose square is ≤ the number in the left-most pair.

Step 3: get the number under the next bar to the right of the remainder. Add up the last digit of the quotient to the divisor. To the right of the sum attained, find an appropriate number which, together with the outcome of the sum, creates a new divisor for the new dividend that is carried down.

Step 4: The new number in the quotient will consist of the same number as chosen in the divisor. The condition is similar— as being either ≤ the dividend.

Step 5: Now, we will use a decimal point and add zeros in pairs to the remainder.

Step 6: The quotient therefore attained is the square root of the number.

Simplify square roots using the prime factorization method works when the given number is a perfect square number.

FAQ (Frequently Asked Questions)

Q1. How to Simplify Square Root When the Radicals are Alike?

Answer: we can add or subtract the numbers in square roots themselves only when the values under the radical sign are equal. Then simply add or subtract the coefficients i.e. the numbers that lie in front of the radical sign and keep intact the original number in the radical sign. For example let’s simplify: √192 using this technique:

√192 = √2 × √96

√96 = √2 × √48

√48 = √4 × √12

√12 = √4 × √3

√192 = √(2 × 2 × 4 × 4) × √3

= √4 × √4 × √4 × √3

= 8√3

Q2. How is Square Root of 50 Simplified?

Answer: Your ultimate solution must be 5 times the square root of 2 which is the simplified version of the square root of 50, which, when simplified, should come equivalent to 7.05. Similarly, we can also perform the square root of 20 simplified as well as the square root of 32 simplified.

Q3. What is the Square Root of 72 Simplified?

Answer: This outcome equals the square root we are trying to simplify. In this example, √72=2⋅3√2 = 8.48528137424.