square root of 225

The square root of any number after exponentiation gives the original value. The square root of 225 is represented using the under root as √225 or it can also be denoted by writing the half power of the number as (225)^1/2.

In this article, we will study the value of the square root of 225 and examine two different methods to calculate the square root value of 225. We will also discuss the nature of the square root of 225.

What is the Square Root of 225?

The square root of 225 is a number that, when multiplied by itself, equals 225.

15 × 15 = 225

-15 × -15 = 225

Therefore, the square root of 225 is either +15 or −15. Note that we can say that there are two values of the square root of a number.

How to Find the Square Root of 225?

The formal definition of the square root of a number is

When the square root of a number is multiplied by itself (i.e. squared), it provides us with the original number. Thus the term ‘square’ was born from the square root.

The method used for calculating the square root may differ depending on the type of number so, firstly we must check if the number is a perfect square or not.

Perfect Squares are the numbers whose roots are integers meaning that the square root is an integer and the original number can be obtained after the multiplication of this integer by itself. For example, 9 is a perfect square because it is obtained by multiplying an integer 3 with itself i.e. 3×3=9.

Dividing the numbers into categories is utterly important to understand the amount of calculations that need to be done for calculating the square root. It is easier to calculate the square root of a perfect square number as compared to a non-perfect square number, this is an obvious observation because the square root of a non-perfect square includes a number followed by decimal numbers that need to be calculated separately but this extra calculation is not needed for perfect squares. Some examples of perfect squares are 4, 9, 16, 25, and 36.

We usually use one of these two methods, in order to find the square root of a perfect square:

• Long Division Method
• Prime Factorization Method

Square Root of 225 by Long Division Method

In this method, we will perform long division to come up with the square root of 225. To compute the square root of 225, we need to follow the steps given below:

Step 1: Represent the number in writing. For example, the number 225 will be written down as 225.00 to understand the integral and decimal parts of the number.

Step 2: In this method, we treat the parts of the number rather than dealing with the whole number to reduce the complexity. For this purpose, we will have to create pairs from the original number. We place bars on the number starting from a unit place, these bars cover two places of the number.

The unit place is considered to begin from the right side of the number. In our case, the number of digits in 225 is odd and we can’t make exact pairs. We will place a bar on digits 25 and the digit 2 to make two pairs.

Step 3: Now, we have to choose a divisor that should satisfy the condition that “its square is less than or equal to the number under the first bar”. In this case, we will find a number whose square is less than 2. So our divisor will be 1. ‘2’ will be the dividend in this case. Divide and write the quotient. Here the quotient is 1 and the remainder is 1.

Step 4: Next, the remainder will be brought and it will be followed by the number under the bar. Here we will bring down ’25’ and since the remainder was ‘1’ therefore our new dividend will be ‘125’.

Step 5: Now we will double the original quotient and write it on the left side and this will be one of our digits of the divisor. In this example, we will double the number ‘1’ to get ‘2’ as shown in the image below. Then, we need to pick a number at the unit place of the divisor which when multiplied to whole divisor is less than or equal to our new dividend.

Here, we select 5 since 25 × 5 is the largest number which is less than equal to dividend 125.

Step 6: The remainder after this calculation is zero as seen below. Since our final remainder is zero, it means we have successfully calculated the square root of our number. The square root will be the value of the quotient.

In our case, the value of the square root is 15.

Long Division Method

The long division method for number 225 has been demonstrated above.

Square Root of 225 by Prime Factorization

Another method for calculating the square root of 225 is the prime factorization method. As the name suggests, this method involves writing down all the prime factors of a number. Let us see how.

Step 1: Take the number whose square root is to be calculated and generate all its prime factors. Prime factors are the numbers whose divisors are 1 and the number itself. For 225, the prime factors will be

225= 3 x 3 x 5 x 5

Step 2: Now, we are required to match the prime factors, the pairs formed have the same numbers. Note that it is only possible if the number has an even number of prime factors. For 225, pairs

225= (3 x 3) x (5 x 5)

Step 3: After generating the pairs, take the square roots on both sides to calculate the square root of the original number. For 225,

√225= √(3 x 3) x √(5 x 5

√225= 3 x 5

√225= 15

Therefore, the square root of 225 is the number 15.

Prime Factors of 225

Note: This method is only useful for Perfect Square and Numbers containing small radicals values.

Is the Square Root of 225 Rational or Irrational?

Let us now see whether the Square Root of 225 is Rational or Irrational. For this, we will have to understand what a rational number means.

Rational Numbers

The formal definition of a Rational number is

A number n is called a rational number, if and only if n can be represented in the form of p/q where q ≠ 0.

From the above definition, we can say that any fraction will satisfy the above condition and therefore all fractions will be rational numbers. Note that the denominator and numerator of fractions must be integers and the denominator must not be equal to zero.

Now we know that the square root of 225 is 25. We can write 25 as

25= 25/1

Hence 25 =p/q, where p=25 and q=1.

Since p and q are integers and q ≠ 0

∴25 is a rational number

FAQs on Square Root of 225

What is the Value of a Square Root of 225?

The square root of 225 is 15 and -15.

Is the Square Root of 225 an Irrational Number?

The square root of 225 is considered a rational number because it can be expressed as a simple fraction or quotient of two integers. This ratio is equal to 15/1.

What is the root 225 symbol?

Root 225, is represented by the square root symbol √ and written as √225. We can also use the half-power representation as 225^1/2.

Can a square root have two values?

Yes, a square root can have two values. This is because squaring either a positive or a negative number results in a positive value.