What is the Square Root of a Number?
The square root of a number is a value that, when multiplied by itself, results in the original number. For example, 6 multiplied by 6 equals 36, so 6 is the square root of 36. Similarly, 4 is the square root of 16, 7 is the square root of 49, 2 is the square root of 4, and so on.
The square root is represented by the symbol “√“. As a result, the square roots of numbers are written as \(\sqrt 4,\sqrt 5,\sqrt 6,\) and so on.
Square Root and the Area of a Square
The word “square” in square root represents the area of a square, which is equal to side times side, and the square root represents the length of one of the sides of the square.
Let us take a square with side length x. The area of the square will be:
\(Area=x~\times~x=x^2\)
So the square root of the area, which is \(\sqrt {Area}=\sqrt{x^2}=x\) (side length)
Tricks to find the Square Root of a Number Easily
A perfect square or a square number is a number whose square root is an integer. The tricks to find square roots are based on the square of the ones place digit. So, to apply the tricks, we need to remember the squares of the ones place digit.
Three-Digit Number Square Root Tricks
The square of a three-digit number is always a two-digit number. Let us take an example of how to find the square root of a three-digit number.
Let us find the square root of 169.
Arrange the digits in pairs from the right hand side – 169
169 has 9 at the ones place. As a result, the square root will have either a 3 or a 7 in the ones place. (As \(3^2=9\) and \(7^2=49\))
When we look at the initial digit, 1, we can see that it is the square of 1. As a result, the tens place digit of the square root will be 1.
Since 1 is the smallest number in the squares, the square root of 169 will have the smaller of the two numbers – 3 and 7, that is, 3. So the tens place digit of the square root is 3.
Hence, the square root of 169 is 13, or \(\sqrt {169}=13\).
Four-Digit Number Square Root Shortcuts
Let us consider an example to understand the trick to find the square of a four-digit number.
Consider finding the square root of 4624.
4624 has 4 in its ones place. As a result, the square root will have either a 2 or an 8 in its ones place.
Now take the first two digits, that is, 46. The number 46 lies between the squares of 6 and 7.
The digit at the tens place of the square root will be the smaller of the two numbers, 6 and 7. So, the square root will have 6 at the tens place.
We can say that the square root will be either 62 or 68.
The tens digit is 6, and the number next to 6 is 7. Take the product of 6 and 7, that is, 6 x 7 = 42. Now 42 < 46.
So, the square root of 4624 will be the bigger of the two numbers, 62 and 68, that is, 68.
Hence, the square root of 4624 is 68, or \(\sqrt {4624}=68\).
Similarly, we can find the square roots of numbers with more digits.
Solved Square Root Shortcuts with Examples
Example 1: Evaluate the given expression
\(6\sqrt{49}~+~8\)
Solution:
\(6\sqrt{49}~+~8\)
The square root of 49 is 7
\(=6~\times~7~+~8\)
Simplify,
= 42 + 8
= 50
Hence, \(6\sqrt{49}~+~8\) = 50.
Example 2: Find the value of \(\frac{1}{9}~+~\sqrt\frac{16}{4}~-~\frac{1}{9}\).
Solution:
\(\frac{1}{9}~+~\sqrt\frac{16}{4}~-~\frac{1}{9}\)
Simplifying the fraction \(\frac{16}{4}\), we get 4.
\(=\frac{1}{9}~+~\sqrt 4~-~\frac{1}{9}\)
The square root of 4 is 2
\(=\frac{1}{9}~+~2~-~\frac{1}{9}\)
Simplifying further, we get,
= 2
Hence, \(\frac{1}{9}~+~\sqrt\frac{16}{4}~-~\frac{1}{9}\) = 2.
Example 3: Solve the equation \(12b^2~-~\sqrt{144}\)
Solution:
\(12b^2~-~\sqrt{144}\)
The square root of 144 is \(\pm~12\),
\(=12b^2~-~(\pm~{12})\)
Dividing 12 throughout the equation, we get
\(=b^2~-~(\pm~1)\)
\(b^2=\pm~1\)
\(b=\sqrt {\pm~1}\)
So \(b=\pm~1\)
Hence, the solution is So \(b=\pm~1\).
The square root of a number is the value that when multiplied by itself results in the original number.
A perfect square is a number whose square root is also an integer. It is also known as the square number.
The square roots of numbers are generally positive integers. However, since the square of a negative number is also positive, negative square roots are possible. For example, the square roots of 4 are 2 and -2, as (-2) x (-2) = 4.
By memorizing the perfect squares and their square root values, we may find the square root by using trial and error. The ones digit of perfect squares helps in swiftly determining the square roots of larger numbers.
The symbol stands for plus or minus. In some cases, there may be a numerical value with both the signs. Hence, this symbol is used to convey that the result may be positive or negative.