 # Square Root of 400 + Solution With Free Steps The square root of 400 is denoted by √400 = 20. The statement can also be written as 400$^{1/2}$ = 20. 400 is the square of 20 as both square and square root are vice versa of each other. The number 400 is a perfect square.

In this article, we will analyze and find the square root of 400 using various mathematical techniques, such as the approximation method and the long division method.

## What Is the Square Root Of 400?

The square root of the number 400 is 20.

The square root can be defined as the quantity that can be doubled to produce the square of that similar quantity. In simple words, it can be explained as:

√400 = √(20 x 20)

√400 = √(20)$^2$

√400= ±20

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining 20. Hence 20 is 400’s square root. The square root generates both positive and negative integers.

## How To Calculate the Square Root of 400?

You can calculate the square root of 400 using any of two vastly used techniques in mathematics; one is the Approximation technique, and the other is the Long Division method.

The symbol √ is interpreted as 400 raised to the power 1/2. So any number, when multiplied by itself, produces its square, and when the square root of any squared number is taken, it produces the actual number.

Let us discuss each of them to understand the concepts better.

### Square Root by Long Division Method

The process of long division is one of the most common methods used to find the square roots of a given number. It is easy to comprehend and provides more reliable and accurate answers. The long division method reduces a multi-digit number to its equal parts.

Learning how to find the square root of a number is easy with the long division method. All you need are five primary operations- divide, multiply, subtract, bring down or raise, then repeat.

Following are the simple steps that must be followed to find the square root of 400 using the long division method:

### Step 1

First, write the given number 400 in the division symbol, as shown in figure 1.

### Step 2

Starting from the right side of the number, divide the number 400 into pairs such as 00 and 4.

### Step 3

Now divide the digit 4 by a number, giving a number either 4 or less than 4. Therefore, in this case, the remainder is zero, whereas the quotient is 2.

### Step 4

After this, bring down the next pair 00. Now the dividend is 00. To find the next divisor, we need to double our quotient obtained before. Doubling 2 gives 4; hence consider it as the next divisor.

### Step 5

Now pair 4 with another number to make a new divisor that results in $\leq$ 00 when multiplied with the divisor. If the number is not a perfect square, add pair of zeros to the right of the number before starting division.

### Step 6

Adding 0 to the divisor and multiplying 40 with 0 results in m $\leq$ 00. The remainder obtained is 0.

### Step 7

The resulting quotient 20 is the square root of 400. Figure 1 given below shows the long division process in detail: ### Important points

• The number 400 is a perfect square.
• The number 400 is a rational number.
• The number 400 can be split into its prime factorization.

## Is Square Root of 400 a Perfect Square?

The number 400 is a perfect square. A number is a perfect square if it splits into two equal parts or identical whole numbers. If a number is a perfect square, it is also rational.

A number expressed in p/q form is called a rational number. All the natural numbers are rational. A square root of a perfect square is a whole number; therefore, a perfect square is a rational number.

A number that is not a perfect square is irrational as it is a decimal number. As far as 400 is concerned, it is a perfect square. It can be proved as below:

Factorization of 400 results in 20 x 20  that can also be expressed as 20$^2$.

Taking the square root of the above expression gives:

= √(20$^20$)

= (20$^2$)$^{1/2}$

= 20

This shows that 400 is a perfect square and a rational number.

Therefore the above discussion proves that the square root of 400  is equivalent to 20. Images/mathematical drawings are created with GeoGebra.