Square Roots of Decimals: The product of prime numbers can be used to express any integer. The prime factorization method represents a number in terms of the product of prime numbers. It is the simplest way for calculating the square root of integers. This technique can be used to find the square root of decimal integers as well by hand. When the number (integer or decimal) involved is large; this procedure becomes repetitive and tiresome. We utilise the division technique to solve this problem.

The value of power \(\frac{1}{2}\) of a number is the square root of that number. A decimal number’s square root is a value that gets the original number when multiplied by itself. The sign for it is \(\sqrt{\text { . }}\) In this article, let’s understand everything about Square Roots of Decimals in detail.

What are Square Roots?

Squaring a number is the inverse procedure of square rooting it. The square root of a number is the number that needs to be multiplied by itself to get the original number, whereas the square of a number is the number that needs to be multiplied by itself to get the original number. If ‘\(x\)’ is equal to the square root of ‘\(y\),’ then \(x \times x=y\).

Every integer has two square roots, one positive value and one negative value because the square of any number is always a positive number. For example, Both \(3\) and \(-3\) are square roots of \(9\). However, you will see that only the positive value is expressed as the square root in most instances.

How to Find Square Roots to Decimals?

Square roots in the decimal format of rational numbers: Remember that the square root of a rational number \(x\) is the rational number \(y\) that produces the number \(x\) when multiplied by itself. That is to say,

\(y^{2}=x \Rightarrow \sqrt{x}=y\)

Example: \((0.4)^{2}=0.16 \Rightarrow \sqrt{0.16}=0.4\) and

\((0.41)^{2}=0.1681 \Longrightarrow \sqrt{0.1681}=0.41\)

The square of a decimal fraction has twice the number of decimal places as the integer itself, as shown by the calculations above. As a result, the number of decimal places in each decimal fraction’s square root is half of the number of decimal places in the provided value.

As a result, the tenth decimal place comes from the first two decimal places to the right, the hundredth decimal place from the third and fourth decimal places, and so on.

Learn The Concept Of Squares And Square Roots

Types of Square Roots of Decimals

Square roots of decimals can be calculated using the long division method:

When calculating the square root of a decimal number using the long division method, the number of digits in a perfect square is significant.

The steps for calculating the square root of a decimal integer in decimal form are as follows:

• Step 1: Get the decimal form of the number. 
• Step 2: As with determining the square root of a perfect square of any integer, place bars on the integral component. 
• Step 3: If necessary, add a zero to the extreme right of the decimal component to make the number of decimal places even.
• Step 4: Starting with the first decimal place, put bars on the decimal component of each pair of digits. 
• Step 5: Begin determining the square root using the long division approach, and as soon as the integral component is exhausted, place the decimal point in the square root. 

The process will be demonstrated using the examples below.

Example: Compute the square root of the decimal number \(477.4225\)

Solution: The instructions below will show you how to use the long division method to calculate the square root of the decimal value \(477.4225\):

Write the decimal number down, then separate the integer and fractional elements into pairs. Then, starting from the commencement of the decimal point, the pair of integers of a decimal number is generated from right to left, and the pair of fractional parts is created from left to right. 

In the decimal number \(477.4225\), for example, \(4\) is one pair, \(77\) is one pair, \(42\) is one pair, and \(25\) is another pair.

Now,

Thus, \(\sqrt{477.4225}=21.85\)

Examples of Square Roots of Decimals

Consider the following approach for determining the decimal number’s square root. For a clear understanding, it is presented with the help of an example.

Example: Calculate the square root of the decimal number \(2.25\).

Solution: The following steps will show you how to use the long division method to calculate the square root of the decimal number \(2.25\):

• Step 1: Write the decimal number down, then separate the integer and fractional elements into pairs. Then, starting from the commencement of the decimal point, the pair of integers of a decimal number is generated from right to left, and the pair of fractional parts are created from left to right. For example, in the decimal number \(2.25,2\) is one pair, and \(25\) is another pair.
• Step 2: Calculate the amount whose pair is less than or equal to the principal pair. \(1\) square is equal to \(1\) in the number \(2.25\). As a result, we will put \(1\) as the divisor and \(1\) as the quotient.
• Step 3: We’ll now subtract \(1\) from \(2\) and get \(1\). The opposite pair, \(25\), will be brought down and the percentage point will be placed within the quotient.
• Step 4: Multiply the divisor by two now. We will write \(2\) below the divisor because \(1\) multiplied by \(2\) equals \(2\). The third digit of the amount must be found for it to be divisible by \(125\).
• Step 5: In the quotient’s place, write \(5\). As a result, the answer is \(1.5\)

That is,

How to Solve Square Roots?

To simplify a square root, we must first determine the number’s prime factorization. If you cannot group a factor, keep it under the square root sign. \(\sqrt{x^{2} y}=\sqrt{(x \times x \times y)}=x \sqrt{y}, w\) here \(x\) and \(y\) are positive integers, is the rule for simplifying square roots.

As an example, \(\sqrt{27}=\sqrt{3 \times 3 \times 3}=3 \sqrt{3}\).

Uses of Square Roots of Decimals

In geometry, the square root of the decimal number is used to find the length of the side if the area of a square is known.

Example: If the area of a square field is \(2.56 \mathrm{~m}^{2}\), then find the length of its side.

Solution: We have, the area of a square field is \(2.56 \mathrm{~m}^{2}\).

Let the length of its side be \(x\).

Area of square \(=x^{2}=2.56 \mathrm{~m}^{2} \Rightarrow x=\sqrt{2.56}=1.6 \mathrm{~m}\)

Hence, the length of the sides of a square field is \(1.6 \mathrm{~m}\).

Solved Examples on Square Roots of Decimals

Q.1. Find the square root of the decimal \(\sqrt{12.25}\).
Ans:

Hence, \(\sqrt{12.25}=3.5\)

Q.2. The area of a square plot is \(23.04 \mathrm{~m}^{2}\). Find the side of the square plot.
Ans: The area of a square plot \(=23.04 \mathrm{~m}^{2}\)
Therefore, the side of the square plot \(=\sqrt{23.04 \mathrm{~m}^{2}}\)
Now,

We find that \(\sqrt{23.04 \mathrm{~m}^{2}}=4.8 \mathrm{~m}\)
Hence, the side of the square plot is \(4.8 \mathrm{~m}\).

Q.3. Find the square root of the decimal number 7.29
Ans:

Hence, \(\sqrt{7.29}=2.7\)

Q.4. Given that \(\sqrt{5}=2.236\), evaluate the following \(\sqrt{\frac{36}{5}}\).
Ans: \(\sqrt{\frac{36}{5}}=\frac{6}{\sqrt{5}}=\frac{6}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}=\frac{6 \sqrt{5}}{5}\)
\(=\frac{6 \times 2.236}{5}=\frac{13.416}{5}=2.6832\)

Q.5. Find the square root of 2 correct to two decimal places.
Ans: Because we need to get the square root of \(2\) to two decimal places, we will start by finding the square root of \(3\) decimal places. This is accomplished by adding \(6\) zeros to the right of the decimal point. As a result, we write \(2=2.000000\).
Now,

Therefore, \(\sqrt{2}=1.414\) upto three decimal places.
\(\Rightarrow \sqrt{2}=1.41\) correct upto two decimal places.
Hence, \(\sqrt{2}=1.41\)

Summary

In this article, we learnt about the definition of square roots, how to find square roots to decimals, types of square roots of decimals, examples of square roots of decimals, how to solve square roots, uses of square roots of decimals, solved examples on square roots of decimals, FAQs on square roots of decimals.

FAQs on Square Roots of Decimals

Q.1. How do you find the square root of a decimal number?
Ans: When calculating the square root of a decimal integer using the long division method, the number of digits in a perfect square is significant. The steps for calculating the square root of a decimal integer in decimal form are as follows:

• Step 1: Simply add a zero to the sharp right of the decimal component to make the overall number of decimal places even (only if required).
• Step 2: Point out the periods inside an integral component, just like we do when we’re trying to identify the root of an ideal square of a number.
• Step 3: Beginning with the first decimal place, highlight the periods on each pair of digits within the decimal component.
• Step 4: Now, using the division method, calculate the square root of the decimal value.
• Step 5: As soon as the integral component is exhausted, insert the percentage point into the root.

Q.2. Define square roots?
Ans: Squaring a number is the inverse procedure of square rooting it. The square root of a number is the number that needs to be multiplied by itself to get the original number, whereas the square of a number is the number that needs to be multiplied by itself to get the original number. If ‘x’ is equal to the square root of ‘y,’ then x times x=y.

Q.3. How do you solve square roots?
Ans: To simplify a square root, we must first determine the number’s prime factorization. If you cannot group a factor, keep it under the square root sign. \(\sqrt{x^{2} y}=\sqrt{(x \times x \times y)}=x \sqrt{y}, y\) here \(x\) and \(y\) are positive integers, is the rule for simplifying square roots.
As an example, \(\sqrt{12}=\sqrt{2 \times 2 \times 3}=2 \sqrt{3}\)

Q.4. What are the methods to find square roots?
Ans: There are three methods to find the square roots. They are
1. Subtraction is used to find the square root.
2. Prime factorisation is a method of determining the square root.
3. Using the division method to find the square root

Q.5. What is the formula of the square root?
Ans: To find the square root of a number, use the square root formula. The exponent formula is as follows: \(\sqrt[n]{x}=x^{\frac{1}{n}}\). We call it square root when \(n=2\). For obtaining the square root, we can use any of the methods listed above, such as prime factorization, long division, and so on.