Square Root
The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number. If 'a' is the square root of 'b', it means that a × a = b. The square of any number is always a positive number, so every number has two square roots, one of a positive value, and one of a negative value. For example, both 2 and -2 are square roots of 4. However, in most places, only the positive value is written as the square root of a number.
What is a Square Root?
The square root of a number is that factor of a number which when multiplied by itself gives the original number. Squares and square roots are special exponents. Consider the number 9. When 3 is multiplied by itself, it gives 9 as the product. This can be written as 3 × 3 or 32. Here, the exponent is 2, and we call it a square. Now when the exponent is 1/2, it refers to the square root of the number. For example, √n = n1/2, where n is a positive integer.
Square Root Definition:
The square root of a number is the value of power 1/2 of that number.
In other words, it is the number whose product by itself gives the original number.
It is represented using the symbol '√ '. The square root symbol is called a radical,
whereas the number under the square root symbol is called the radicand.
Let us say m is a positive integer, such that √(m.m) = √(m2) = m
In mathematics, a square root function is defined as a one-to-one
function that takes a positive number as an input and returns the
square root of the given input number.
f(x) = √x
For example, if x=4, then the function returns the output value as 2.
Note: The square root of a negative number represents a complex
number.
Suppose √-n = i√n, where i is the imaginary number.
Square Root Formula:
The formula to find the square root is:
y = √a
Since, y.y = y2 = a; where ‘a’ is the square of a number ‘y’.
Properties of Square root
Here, some of the linear algebra applications are given as:
- If a number is a perfect square number, then there exists a perfect square root.
- If a number ends with an even number of zeros (0’s), then it can have a square root.
- The two square root values can be multiplied. For example, √3 can be multiplied by √2, then the result should be √6.
- When two same square roots are multiplied, then the result should be a radical number. It means that the result is a non-square root number. For instance, when √7 is multiplied by √7, the result obtained is 7.
- The square root of any negative numbers is not defined because the perfect square cannot be negative.
- If a number ends with 2, 3, 7 or 8 (in the unit digit), then the perfect square root does not exist.
- If a number ends with 1, 4, 5, 6 or 9 in the unit digit, then the number may have a perfect square root.
How to Find Square Root?
To find the square root of a number, we just see by squaring which number would give the actual number. It is very easy to find the square root of a number that is a perfect square. Perfect squares are those positive numbers that can be expressed as the product of a number by itself. In other words, perfect squares are numbers which are expressed as the value of power 2 of any integer. We can use four methods to find the square root of numbers and those methods are as follows:
- Repeated Subtraction Method
- Prime Factorization Method
- Estimation Method
- Long Division Method
It should be noted that the first three methods can be conveniently used for perfect squares, while the fourth method, i.e., the long division method can be used for any number whether it is a perfect square or not.
Repeated Subtraction Method of Square Root:
This is a very simple method. We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of 16 using this method.
- 16 - 1 = 15
- 15 - 3 = 12
- 12 - 5 = 7
- 7 - 7 = 0
Square Root by Prime Factorization Method:
Prime factorization of any number means to represent that number as a product of prime numbers.
To find the square root of a given number through the prime factorization method, we follow
the steps given below:
- Step 1: Divide the given number into its prime factors.
- Step 2: Form pairs of factors such that both factors in each pair are equal.
- Step 3: Take one factor from the pair.
- Step 4: Find 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 us find the square root of 144 by this method.
This method works when the given number is a perfect square number.
Finding Square Root by Estimation Method:
Estimation and approximation refer to a reasonable guess of the actual value to make
calculations easier and more realistic. This method helps in estimating and approximating
the square root of a given number. Let us use this method to find √15. Find the nearest
perfect square number to 15. 9 and 16 are the perfect square numbers nearest to 15.
We know that √16 = 4 and √9 = 3. This implies that √15 lies between 3 and 4.
Now, we need to see if √15 is closer to 3 or 4. Let us consider 3.5 and 4.
Since 3.52 = 12.25 and 42= 16. Thus, √15 lies between 3.5 and 4 and is closer to 4.
Let us find the squares of 3.8 and 3.9. Since 3.82 = 14.44 and 3.92 = 15.21.
This implies that √15 lies between 3.8 and 3.9. We can repeat the process and check
between 3.85 and 3.9. We can observe that √15 = 3.872.
This is a very long process and time-consuming.
Calculating Square Root by Long Division Method:
Long division is a method for dividing large numbers into steps or parts, breaking the
division problem into a sequence of easier steps. We can find the exact square root of
any given number using this method. Let us understand the process of finding square root
by the long division method with an example. Let us find the square root of 180.
- Step 1: Place a bar over every pair of digits of the number starting from the units' place (right-most side). We will have two pairs, i.e., 1 and 80
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Step 2: We divide the left-most number by the largest number whose square is less than or equal
to the number in the left-most pair.
- Step 3: Bring down the number under the next bar to the right of the remainder. Add the last digit of
the quotient to the divisor. To the right of the obtained sum, find a suitable number which,
together with the result of the sum, forms a new divisor for the new dividend that is carried down.
- Step 4: The new number in the quotient will have the same number as selected in the divisor. The condition is the same — as being either less than or equal to the dividend.
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Step 5: Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder.
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Step 6: The quotient thus obtained will be the square root of the number. Here, the square root of 180 is
approximately equal to 13.4 and more digits after the decimal point can be obtained by repeating the same process as follows.
Square Root Table
The square root table consists of numbers and their square roots. It is useful to find the squares of numbers as well. Here is the list of square roots of perfect square numbers and some non-perfect square numbers from 1 to 10.
Square Root of Decimal:
A decimal value will have a dot (.) such as 3.8, 5.2, 6.33, etc. For a whole number, we have understood how to derive the square root but let us see how to get the square root of a decimal.
T: V → W
Example: Find the square root of 0.09.Let N2 = 0.09
Taking root on both sides.
N = ±√0.09
As we know,
0.3 x 0.3 = (0.3)2 = 0.09
Therefore,
N = ±√(0.3)2
N = ±(0.3)
Square Root of Negative Number:
The square root of a negative number cannot be a real number, since a square is either a positive number or zero. But complex numbers have the solutions to the square root of a negative number. The principal square root of -x is: √(-x)= i√x. Here, i is the square root of -1. For example: Take a perfect square number like 16. Now, let's see the square root of -16. There is no real square root of -16. √(-16)= √16 × √(-1) = 4i (as, √(-1)= i), where 'i' is represented as the square root of -1. So, 4i is a square root of -16.
Square of a Number:
Any number raised to exponent two (y2) is called the square of the base. So, 52 or 25 is referred to as the square of 5, while 82 or 64 is referred to as the square of 8. We can easily find the square of a number by multiplying the number two times. For example, 52 = 5 × 5 = 25, and 82 = 8 × 8 = 64. When we find the square of a whole number, the resultant number is a perfect square. Some of the perfect squares we have are 4, 9, 16, 25, 36, 49, 64, and so on. The square of a number is always a positive number.
How to Find the Square of a Number?
The square of a number can be found by multiplying a number by itself. For single-digit numbers, we can use multiplication tables to find the square, while in the case of two or more than two-digit numbers, we perform multiplication of the number by itself to get the answer. For example, 9 × 9 = 81, where 81 is the square of 9. Similarly, 3 × 3 = 9, where 9 is the square of 3.
The square of a number is written by raising the exponent to 2. For example,
the square of 3 is written as 32 and is read as "3 squared".
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Here are some examples:
- 42 = 4 × 4 = 16
- (-6)2 = -6 × -6 = 36
- (5/3)2 = 5/3 × 5/3 = 25/9
Squares and Square Roots:
There is very strong relation between squares and square roots as each one of them is the inverse relation of the other. i.e., if x2 = y then x = √y.
It can be simply remembered like this:
- When "square" is removed from one side of the equation, we get the square root on the other side. For example, 42 = 16 means, 4 = √16. This is also known as "taking square root on both sides".
- When "square root" is removed from one side of the equation, we get square on the other side. For example, √25 = 5 means, 25 = 52. This is also known as "squaring on both sides"
Example: Solve the equation √(2x + 3) = 10.
Solution:
Squaring on both sides of the equation would result in the cancellation of
the square root on the left side. 2x + 3 = 102
2x + 3 = 100
2x = 97
x = 97/2 = 48.5