Square Root of 400
Phil's father tells him that their square shed is 400 sq. yards in area. Phil wonders what could be the length of the sides of the shed. He takes the measuring tape to measure the sides. To find the area, we have to find the square of the sides (side x side). In order to find the side of the square when the area is given, we need to find the square root. We know that 20 × 20 = 400. Thus, the length of the sides of the shed is 20 yards.
In this lesson, we will calculate the square root of 400 by long division method along with a few interesting problems.
 Square Root of 400: 20
 Square of 400: 1,60,000
1.  What Is the Square Root of 400? 
2.  Is Square Root of 400 Rational or Irrational? 
3.  How to Find the Square Root of 400? 
4.  FAQs on Square Root of 400 
What Is the Square Root of 400?
 The square root of a number is the number that when multiplied to itself gives the original number as the product.
 400 = a × a = 20^{2}
 Then a = √400 = √(20 × 20)
 20 × 20 = 400 or 20 × 20 = 400
 The 2nd root of 400 is +20 or  20
 This shows that 400 is a perfect square.
Is the Square Root of 400 Rational or Irrational?
A number that can be expressed as the ratio of two integers, that is, p/q where q is not equal to 0, is called a rational number. Now, let us look at the square root of 400.
√400 = 20 = 20/1. Thus, the square root of 400 is a rational number.
How to Find the Square Root of 400?
The square root of 400 can be calculated using methods such as: Prime Factorization or Long Division method or Repeated Subtraction Method.
Square Root of 400 by Repeated Subtraction Method
Start from 400 and keep subtracting successive odd numbers till we obtain zero. The total numbers we subtract is the square root of 400.
 400  1 = 399
 399  3 = 396
 396  5 = 391
 391  7 = 384
 384  9 = 375
 375  11 = 364
 364  13 = 351
 351  15 = 336
 336  17 = 319
 319 19 = 300
 300  21 = 279
 279  23 = 256
 256  25 = 231
 231  27 = 204
 204  29 = 175
 175  31 =144
 144  33 =111
 111  35 = 76
 76  37 = 39
 39  39 = 0
Thus, starting from 400, we have subtracted 20 times to obtain 0. Thus, the square root of 200 is 20.
Square Root of 400 by Long Division Method
Let us follow these steps to find the square root of 400 by long division.
 Step 1: Group the digits into pairs (For digits to the left of the decimal point, pair them from right to left) by placing a bar over it. Since our number is 400, let us represent it as inside the division symbol.
 Step 2: Find the largest number such that when you multiply it with itself, the product is less than or equal to 4. We know that 2 × 2 = 4. Now, let us divide 4 by 2.
 Step 3: Bring down the next pair of numbers, which is, 00. Multiply the quotient 2 by 2 and write it in the new divisor's place. Here, it is 4.
 Step 4: Choose a number in the unit's place for the new divisor such that its product with a number is less than or equal to 0. We know that 4 is in the ten's place and our product has to be 0, which means, 40 × 0 = 0. The complete process of the long division stops here as the remainder is 0. Thus, the quotient 20 is the square root of 400.
Explore square roots using illustrations and interactive examples.
Think Tank
 Are all square roots rational?
 What are numbers with integer square roots called?
 Can square roots be negative?
Important Notes
 The square root of 400 in the radical form is expressed as √400
 In exponent form, the square root of 400 is expressed as 400^{1/2}
 The real roots of √400 are +20 or 20.
Square Root of 400 Solved Examples

Example 1: Mark wants to fence his square backyard. The area of his backyard is 400 square feet. What is the length of fence wire Mark will need?
Solution:
To fence his backyard, Mark needs to know the length of each side. All the sides of the yard are equal as it is a square backyard. Therefore, we need to determine the square root of 400.
20 × 20 = 400
Each side will require 20 feet of fence wire. Thus, he will need 4 × 20 = 80 feet long fence wire.

Example 2: James wants to purchase a new rug for his dining room. At the store, he finds a square rug that has an area of 100 sq. feet.
a. How long is each side of the rug?
b. How many of those rugs are needed to cover an area of 400 square feet?Solution:
Area of one rug = 100 square feet
The length of each side of the rug is √area = √100
The square root of 100 is 10.
Therefore, the length of each side of the rug is 10 feet.
To cover an area of 400 square feet, he needs 400 ÷ 100 = 4 rugs
Thus, each side of the rug is 10 feet long and he needs 4 rugs to cover an area of 400 sq feet.

Example 3: Help Emily determine the value of a if a^{2 }= 400.
Solution:
The square root of 400 can be determined as, √400 = a × a
Using the known multiplication fact, 20 × 20 = 400 and 20 × 20 = 400
a = + 20 or 20
Interactive Questions
FAQs on Square Root of 400
What is the square root of 400?
The square root of 400 is +20 or 20.
Why is the square root of 400 negative 20?
400 = 20 × 20. Thus, the square root of 400 is negative 20.
Is 200 an irrational number?
No, the square root of 400 is a rational number. It can be expressed as 20/1.
What are the methods to find the square root of 400?
We can find the square root of 400 using any of these 3 ways: prime factorization method, long division method, or repeated subtraction method.