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

• The square root of a number is the number that when multiplied to itself gives the original number as the product. 
• 400 = a × a = 20²
• 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.

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.

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.

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.

• Square root of 225
• Square root of 169
• Square root of 125  
• Square root of 100
• Square root of 1000 

• Are all square roots rational?
• What are numbers with integer square roots called?
• Can square roots be negative?

• 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.

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.