Estimating Square Root: Definition, Examples, Methods

Squaring a number is the inverse procedure of square rooting it. Estimating the square root of a number is multiplying by itself to get the original number. For example, to estimate sqrt(6), consider that 6 is between the perfect squares 4 and 9. Sqrt(4) equals 2, and sqrt(9) equals 3. We’d expect 6’s square root to be closer to 2 than to 3 because it’s closer to 4 than to 9.

We commonly use the method of tedious decimal work. In this article, we shall learn how to find square roots or their approximations. The square root of a number is the one that produces the number when multiplied by itself.

What is Square Root?

The inverse process of squaring an integer is the square root. The square root of a number is the number that must be multiplied by itself to get the original number, whereas the square of a number is multiplying a number by itself. If \(a\) is the square root of \(b\), then \(a \times a=b\). The square root of any positive integer is not always a positive number. Every number has two square roots, one positive and one negative.

A number’s square root is the number that is multiplied by itself to produce the product. Exponents are something we’ve learnt about. Special exponents are squares and square roots. Consider the number five. When \(5\) is multiplied by itself, the result is \(25\). A square is a number with an exponent of two. It’s called a square root when the exponent is \(\frac{1}{2}\).
For example, \((n \times n)=n^{2}\), where \(n\) is a positive integer.

Square Root Definition

The square root of a number \(x\) is the number that gives the result \(x\) when multiplied by itself. The square root of \(x\) is denoted as \(\sqrt{x}\). The symbol \(\sqrt{ }\), it means ‘square root of’.

Examples: \(\sqrt{36}=6, \sqrt{144}=12, \sqrt{196}=14\)

Formula to Find the Square Root

The square root is equal to \(\frac{1}{2}\) of the exponent. To find the square root of a number, use the square root formula. We are familiar with the exponent formula: \(\sqrt[n]{a}=a^{\frac{1}{n}}\).

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 factorisation, long division, and so on. Example:
\(9^{\frac{1}{2}}=\sqrt{9}=3\)

Methods to Find Square Root

Different methods to find the square root of a number are discussed below:

1. Prime Factorisation Method

A perfect square’s prime factors can be paired. To find the square root, we must first make prime factor pairs. The product of these factors is then obtained by taking one factor by each pair. The method to find the square root of a perfect square.

Example: Find the square root of \(4225\).

Factorising \(4225\) by the division method, we can write \(4225\) as a product of its prime factors as
\(4225=5 \times 5 \times 13 \times 13\)
So, \(\sqrt{4225}=5 \times 13=65\)
So, the square root of \(4225\) is \(65\).

2. Pattern Method for Finding Square Root Ending with 25

Consider an example \(9025\) to find its square root.

1. As the perfect square ends in \(25\), its square root must end in \(5\). Write \(5\) in the units place.
2. Find two single-digit number numbers whose product is \(90\). So, we have \(9 \times 10=90\). Now we need to consider the smaller number between those two numbers \(9\) and \(10\). In this case, it is \(9\).
3. \(9\) must be present in the tens; hence \(\sqrt{9025}=95\).

3. Finding Square Root of Fractions

Example: Find the square root of the fraction \(\frac{486}{1014}\)

The numerator and the denominator are not perfect squares in the given fraction. But look at the fraction, we observe that they can be further reduced to its lower equivalent fraction. So, if both the numbers are divided by \(6\) (the common factor), both the numbers become perfect squares.

Therefore, \(\frac{486}{1014}=\frac{81}{169}\)

So, \(\sqrt{\frac{486}{1014}}=\sqrt{\frac{81}{169}}=\frac{\sqrt{81}}{\sqrt{169}}=\frac{9}{13}\).

4. Estimation Method to Find the Square Root

Estimation and approximation refer to an acceptable approximation of the actual number. This method helps in the calculation and estimation of a number’s square root.

Example: Let’s see if we can find a square root of \(15\) using this strategy. Find the perfect square numbers that are closest to \(15\).

The perfect square numbers closest to \(15\) are \(9\) and \(16\). We know that \(\sqrt{16}=4\) and \(\sqrt{9}=3\). This means that the square root of \(15\) is lies between \(3\) and \(4\). Now we must determine whether the square root of \(15\) is closer to \(3\) or \(4\).

Consider the numbers \(3.5\) and \(4\).
\(\left(3.5^{2}\right)=12.25\) and \(\left(4^{2}\right)=16\)

As a result, \(\sqrt{15}\) is near to \(4\) and lies between \(3.5\) and \(4\).

We shall proceed this way to get closer and closer to the square root of \(15\) to the desired number of digits after the decimal point.

5. Finding Square Roots of Decimal Numbers

Remember that the square root of a decimal number will have half the number of decimal places as the number itself.

Examples:
1. \(\quad \sqrt{0.0004}=0.02\) since \(0.02 \times 0.02=0.0004\)
2. \(\sqrt{0.81}=0.9\) since \(0.9 \times 0.9=0.81\)

In each case, the decimal part of the number has double the number of decimal places than its square root.

6. Square Root of Decimal by Long Division Method

Let us consider the decimal number \(7624.7824\) to find the square root.

The following steps are used to find the square root:

1. From right to left, pair the numbers before the decimal point.
2. \(76\) and \(24\) are the two periods. There is no need to put \(0\) because the number of decimal places is even.
3. From left to right, pair the numbers after the decimal point.
4. \(78\) and \(24\) are the two periods. As shown, we can use the long division method to calculate the square root.

So, the square root of \(7624.7824\) is \(87.32\).

7. Estimating Square Roots of Non-perfect Squares

An irrational number would be the square root of a number that is not a perfect square. An irrational number is one that cannot be represented as a fraction. These numbers extend continuously in decimal form, without any repeated digit pattern. We can use the long division method to estimate the square root to the desired number of digits after the decimal point.

Example: \(\sqrt{3}=1.732\)

Solved Examples

Q.1. Find the square root of \(0.0144\).
Ans: The square root of \(0.0144\) will have \(2\) decimal places.
The square root of \(144\) is \(12\).
Therefore, the square root of \(0.0144=0.12\).

Q.2. Find the square root of \(\frac{81}{100}\).
Ans: The given fraction is \(\frac{81}{100}\)
The square root of \(81=\sqrt{81}=9\)
The square root of \(100=\sqrt{100}=10\)
So, the square root of \(\frac{81}{100}=\sqrt{\frac{81}{100}}=\frac{9}{10}\)
Therefore, \(\sqrt{\frac{81}{100}}=\frac{9}{10}\).

Q.3. Estimate \(\sqrt{160}\) to the nearest tenth.
Ans:
1. First find the perfect squares nearest to \(160\).
\(144<160<169\)
\(\Rightarrow \sqrt{144}<\sqrt{160}<\sqrt{169}\)
2. The number will be between \(12\) and \(13\).
3. Therefore the whole number part of the answer is \(12\).
4. \((12.5)^{5}=156.25\)
Therefore, as a result, \(\sqrt{160}\) is near to \(13\) and lies between \(12.5\) and \(13\). We may repeat the process to obtain a closer answer.

Q.4. Find the square root of \(2025\).
Ans: We can find the square root of \(2025\) by the division method.

\(2025=5 \times 5 \times 3 \times 3 \times 3 \times 3\)
So, \(\sqrt{2025}=5 \times 3 \times 3=45\)
Therefore, the square root of \(2025\) is \(45\).

Q.5. Find the square root of \(3025\).
Ans: The square root of the last two digits \(25\) is \(5\). And \(30=5 \times 6\). Between the two factors of \(30\), the number \(5\) is the smaller number.
Hence, the square root of \(3025\) is \(55\).

Summary

The inverse process of squaring an integer is the square root. The square root of a number is the number that must be multiplied by itself to get the original number. This article includes the definition of square root, examples, different methods to find the square root of numbers and problems.

This article “estimating square root” help in understanding these in detail, and it helps to solve the problems based on these very easily.

Learn About Properties of Square Numbers

FAQs

Q.1. How to estimate square roots to the nearest tenth?
Ans: We will explain with the help of one example
Example: Estimate \(\sqrt{181}\) to the nearest tenth.
1. First find the perfect squares nearest to \(181\).
\(169<181<196\)
\(\Rightarrow \sqrt{169}<\sqrt{181}<\sqrt{196}\)
2. The number will be between \(13\) and \(14\).
3. Therefore, the whole number part of the answer is \(13\).
4. \((13.25)^{2}=175.56\) and \((13.5)^{2}=182.25\)
5. So, \((13.4)^{2}=179.56\)
Therefore, as a result, \(\sqrt{181}\) is near to \(13\) and lies between \(13.4\) and \(14\).

Q.2. What is a square root?
Ans: The square root of a number \(x\) is the number that gives the result \(x\) when multiplied by itself. The square root of \(x\) is denoted as \(\sqrt{x}\). The symbol \(\sqrt{ }\) it means ‘square root of’.
Examples: \(\sqrt{9}=3, \sqrt{169}=13, \sqrt{225}=15\)

Q.3. How to estimate the square root of a number?
Ans: We can find the square root of a given number by using one example
Let us find the perfect square numbers that are closest to \(15\).
The perfect square numbers closest to \(15\) are \(9\) and\(16\) .
We know that \(\sqrt{16}=4\) and \(\sqrt{9}=3\). This means that \(15\) is lies between \(3\) and \(4\). Now we must determine whether \(15\) is closer to \(3\) or \(4\).
Consider the numbers \(3.5\) and \(4.\) \(\left(3.5^{2}\right)=12.25\) and \(\left(4^{2}\right)=16\)
As a result, \(\sqrt{15}\) is near to \(4\) and lies between \(3.5\) and \(4\).

Q.4. How to calculate estimation in the square root?
Ans: We can find the square root of a given number by using one example
Let us consider an example to find the square root of \(5\).
Step 1: Make an estimate. \(2\) and \(3\) have square numbers of \(4\) and \(9\), respectively. Between these two numbers is the square root of the number \(5\).
Step 2: Make a division. Divide \(5\) by two or three. Let’s go with option \(2\) in this case. We’re going to get \(2.5\).
Step 3: Calculate the average. \(2.25\) is the result of averaging \(2.5\) and \(2\).
Step 4: Do it again. Continue repeating steps \(2\) and \(3\) to get a more precise number. In that situation, we’d multiply \(5\) by \(2.25\), which equals \(2.2.\) So, the square root of \(5\) is estimated to be equal to \(2.2\).

Q.5. What is the \(\sqrt{ }\) called?
Ans: Square root of a number is represented by the symbol \(\sqrt{ }\) radicals.

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