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Suppose you are asked to find the square root of a number. But for that, you need to know what are squares and square roots. The value of the number being multiplied by itself gives the original given number back, known as its square root. For example, let's assume y is the square root of x, then y2 = x. This equation can also be written as y = √x. The square root of a number is represented by the symbol '√ ' Whenever we multiply a number by itself, the positive value we get is a square of the number. We get the original number back by finding out the square root of this positive value.

The square of 4 is 16 or 42 = 16 and √16 = 4. It is easy to find the square root of 16 or such numbers because they are perfect squares. But for the numbers like 10, 7, 20, 24, and many more, it becomes a bit difficult and calculative. There is no such formula to find the square root of a number. But there are various methods and tricks by which calculation of square roots can be easy. In different sections, we will learn about them.

The symbol used to represent the square root is known as the radical symbol. '√' is the square root symbol. If x is the original number, then √x is the square root of the number. Here x is also known as the radicand. Radical and square root both represent the same thing. The square root formula can be given by:

X = √Y, where X is the square root of Y. On the other hand, X2 = Y. Therefore, where Y is the square of X.

In Math, the square root function takes a positive value as input and gives the input value's square root as the output. Hence, it is said to be a one to one function. For example,

F(a) = √a.

Let us consider if x = 25, then the value returned will be 5. Below we have some of the properties of squares and square roots.

A number can have a square root if it ends with an even number of zeros.

A number can be a perfect square. Then, there will be a perfect square root also.

Multiplication of two square root values is possible. For example, √2 × √3 = √6.

If you find square root of a negative number, you get a complex number as a result. Perfect squares are never negative.

We get a radical number by multiplying two same square roots. If you multiply √7 × √7, you get 7 back, which is a nonsquare root number.

Numbers ending with 1, 4, 5, and 6 will have a square root.

Perfect square root does not exist for numbers ending with 2, 3, 7, and 8.

We need to find the square root of a number, whether it's a perfect or imperfect square. You already know if it's an ideal square, then we can use the prime factorization method to factorize and find the square root easily. For other numbers, we have the division method.

It's one of the easiest ways to find the square root of a number. For example, let's take the number 256. Now break 256 into its prime factors. Therefore 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Now we need to pair in doubles of similar numbers and consider them once only, (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2). Hence, √256 = (2 × 2 × 2 × 2) = 16.

It is the square root formula used for imperfect squares or large numbers. Let's consider number 3. It is not a perfect square number. Taking this example, we have illustrated the long division method below.

Writing 3 with a decimal point and groups of two zeros for the decimal. Putting all of this under the long square root symbol (which looks like division), and we start dividing.

1√3.00 00 00 00 00

Putting the largest number whose square is less than or equal to 3 and above the 3. Square it and subtract it from 3, which in this case is 1.

Bringing down a pair of zeros and multiplying the answer so far by two. The next digit in the answer will be put at the end of the doubled answer and multiply that by the new digit. This product is subtracted, and two more zeros are brought down. We have to continue this process for as many digits of accuracy as we want.

The square root of 3 results in 1.732.

Now let us consider the number 24.

Give Some Applications of the Square Root Formula.

Answer: The square root formula is used in geometry, calculus, algebra, quadratic equations, computing, and other mathematical applications.

Give the Square Roots of 5, 7, 9, and 11.

Answer: The square roots of 5, 7, 9, and 11 are 2.236, 2.646, 3.0, and 3.317, respectively.

FAQ (Frequently Asked Questions)

1. What are perfect squares?

Finding square root with perfect square numbers is really very easy. Below we have provided numbers which are perfect squares in nature:

1^{2} = 1, 2^{2} = 4, 3^{2} = 9, 4^{2} = 16, 5^{2} = 25, 6^{2} = 36, 7^{2} = 49, 8^{2} = 64, 9^{2} = 81, 10^{2} = 100.

Hence, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 other numbers are perfect squares in nature. So, if you are giving the number 49, you already know the square root of 49 is 7. But for other numbers, the calculation of square roots becomes a bit tedious.

2. What is the square root equation?

Firstly, isolate or bring the square to any side, iron on the left-hand or right-hand side. Now to solve this square root equation, we need to follow the below steps. The next step is to square both the right-hand and left-hand sides of the equation. Now it becomes easy, and you can see all the rest of the equation. Consider an example, √ (2x + 7) – 10 = 0. Now first, you need to bring the square root term on one side. √ (2x + 7) = 10. The next step is to square both sides.

=> 2x + 7 = 10^{2}

=> 2x + 7 = 100

=> 2x = 93

=> x = 93/2

=> x = 46.5

Hence the value of x is 46.5.