# Square root of 400

Square root of 400 is a value that can be squared to get the original number. Suppose x is the root of 400, then x2 = 400. Basically, a square root is the inverse process of squaring a number. Since, 400 is a perfect square, therefore it is easy to find the square root value.

The square root of 400 is represented by √400, where ‘√’ is the radical symbol and the value under the root is the radicand. √400 is 20, which is a rational number. To find the square root, we can use the simple prime factorisation method or long division method.

 Square root of 400 = √400 = ±20 Or In Exponent Form, (400)½ = ±20

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## How to Find the Square root of 400?

Square root of 400 is a rational number that can be represented in the form of a numerator and denominator.

There are three methods to find the square root of 400, they are:

### Prime Factorisation Method

This method is mostly recommended when we need to find the square root of perfect squares. In this method, we find the prime factors form of 400 and then take the square root on both the sides.

The prime factorisation of 400 is:

400 = 2 x 2 x 2 x 2 x 5 x 5

If we write in exponential form, then;

400 = 24 x 52

Or

400 = 22 x 22 x 52 [Using laws of exponents]

Taking square root on both sides of the equation, we get;

$\sqrt{400} = \sqrt{2^{2}\times 2^{2}\times 5^{2}}$

If we separate the roots for each term, then;

$\sqrt{400} = \sqrt{2^{2}}\times \sqrt{2^{2}}\times \sqrt{5^{2}}$

Now, we can cancel each of the square root on the right hand side of the equation, with the square of each term. Therefore, we get;

√400 = 2 x 2 x 5

√400 = 20

Hence, we got the square root of the rational number 400, which is also a rational number.

 Facts: Square root of 400 is a natural or a whole number 400 is an even number and so is it’s square root 400 is evenly divisible by its square root value, thus, 20 is the factor of 400. The roots of 400 are +20 or -20. 400 sq.units is the area of a square that has side length equal to 20 units.

### Long Division Method

Finding the square root of a number, using long division, is the fastest method. This method is most efficient when we find the square root of imperfect squares because it is difficult to find the square root of imperfect numbers using prime factorisation. Square root of imperfect numbers results in decimal value. Long division method is also useful to find the root of large numbers.

Let us find here the square root of 400 using a long division method.

Step 1: Group the digits of the original number in pairs from right to left and put the bar over each pair. Thus, 4 and 00 are two groups here.

Step 2: Now we will begin with divisions. Take the greatest number, which can be multiplied by itself to get a value equal to 4, such that 4 gets subtracted from 4. Hence, we get 0.

Step 3: Write the next pair of numbers, i.e., 00, on the dividend side and add 2 on the left side, such that 2 + 2 = 4 is the new divisor.

Step 4: Now we need to attach 4 (on the left side) with such a number such that when multiplying the divisor with the same number, the resulting values cancel the complete dividend. Hence, 40 multiplied by 0 results in 00.

The whole division process is given below:

Therefore, the square root of 400 is equal to 20, as per the long division method.

### Repeated Subtraction Method

This method is also applicable only for perfect squares. In this method, we start subtracting the original number by the set of odd numbers, in a successive manner, such that at the end we get zero. The number of times we subtract with the odd numbers is equal to the required square root. For example, if we subtract the number 5 times then the square root of the number is 5. Let us see how it works.

• 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

Therefore, we can see, the subtraction method is done for 20 times. Only after the 20th time, we got the difference equal to zero. Therefore, the square root of 400 is 20.

### Square Roots of Perfect Numbers

• Square Root of 25 = 5
• Square root of 49 = 7
• Square root of 81 = 9
• Square root of 100 = 10
• Square root of 169 = 13
• Square root of 225 = 15
• Square root of 289 = 17

## Solved Examples

Q.1: What is the value of 10 multiplied by √400?

Solution: The value of 10 multiplied by √400 is:

⇒ 10 x √400

⇒ 10 x 20

⇒ 200

Hence, the answer is 200.

Q.2: If the area of a square shaped field is 400 m2, what is the value of its length and breadth?

Solution: Given that,

Area of the field = 400 m2

Area of the square = Side2

Since the field is in square shape, therefore the length and breadth of the field will be equal. Therefore,

Area of the field = Length2

400 = Length2

Or

Length = √400 = 20

Hence, the value of length and breadth of the field is 20 meter, respectively.

Q.3: If the area of square is 400 units, then find the perimeter of the square.

Solution: Given,

Area of the square = 400 units

Therefore,

Side of the square = √area = √400 = 20 unit

Therefore, the perimeter of the square is;

Perimeter = 4side = 4 x 20 = 80 units.

### Practice Questions

1. √400 + √100 + √25 = ?
2. You have a square blanket of 400 sq.inches. What is the length of one side of the blanket?
3. A square table has 20 cm of length. What is the area of the table?
4. The area of the room is 400 square feet. What is the perimeter of the room?

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## Frequently Asked Questions on Square root of 400

### What is the square root of 400?

The square root of 400 is equal to 20.
√400 = 20

### Is the square root of 400 rational or irrational?

The square root of 400 is a rational number because it can be represented in the form of a ratio or p/q.
√400 = 20 = 20/1
Where 20 is the numerator and 1 is the denominator.

### Is 400 a perfect square?

Yes, 400 is a perfect square because the square of the real number 20, results in the actual number.
202 = 400

### Which is the fastest method to find the square root of 400?

Since 400 is a perfect square and only a three digit number, therefore we can use a long division method to find the square root, quickly.

### What is the value of the square root of 4000?

The square root of 4000 is an irrational number, since it is not a perfect square.
√4000 = 20√10
If we further simplify the above value, we will get the decimal value which is non-terminating and non-repeating.
√4000 = 63.2455532033676