# Square & Square Root of 400

Last Updated: April 25, 2024

## Square of 400

400² (400 × 400) = 160,000

To calculate the square of 400, you multiply 400 by itself:

400 × 400 = 160,000.

Therefore, the square of 400 is 160,000. Understanding how to find the square numbers is crucial in various mathematical fields including algebra, geometry, and data analysis, as it provides a foundation for tackling more intricate problems and concepts.

## Square Root of 400

√400 = 20

Thus, the square root of 400 is exactly 20. This calculation is important in various mathematical and practical contexts, such as geometry, where it can represent the side length of a square with an area of 400 square units. It’s also significant in algebra and engineering, where understanding square roots helps in solving problems involving areas and other applications that require precise measurements.

Square Root of 400 : 20

Exponential Form : 400^½ or 400^0.5

## Is the Square Root of 400 Rational or Irrational?

The square root of 400 is rational.

This is because 400 is a perfect square, being 20 × 20 = 400. As a result, the square root of 400 is exactly 20, which can be expressed as a fraction 20/1, making it a rational number.

## Methods to Find Value of Root 400

There are several methods to find the value of the square root of 400:

Prime Factorization Method:

Decompose 400 into its prime factors, which are 2 × 2 × 2 × 5 × 5. Pair up identical factors and take one from each pair to get 2 × 5 = 10. Therefore, the square root of 400 is 20.

Repeated Subtraction Method:

Start subtracting consecutive odd numbers from 400 until you reach 0. Count how many times you subtracted, and that count is the square root. For 400, this method will result in 20 subtractions, indicating the square root is 20.

Estimation Method:

Since 20 is a known square root close to 400, you can estimate by finding a number close to 20 that squares to 400. By trial and error, you can quickly determine that 20 is indeed the square root of 400.

Calculator: Use a calculator with a square root function to directly find the square root of 400, which is 20.

These methods offer different approaches to find the square root of 400, catering to various preferences and situations.

## Square Root of 400 by Long Division Method

Let’s follow these steps to find the square root of 400 by long division:

Step 1: Group the digits into pairs by placing a bar over them. Since our number is 400, represent it inside the division symbol.

Step 2: Find the largest number such that when multiplied by itself, the product is less than or equal to 4. We know that 2 × 2 = 4. Now, 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, resulting in 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. Since 4 is in the ten’s place and our product has to be 0, it means 40 × 0 = 0. The long division process stops here as the remainder is 0. Thus, the quotient 20 is the square root of 400.

By following these steps, we have successfully found that the square root of 400 is 20 using the long division method.

## 400 is Perfect Square Root or Not

Yes, 400 is a perfect square.

A perfect square is a number that can be expressed as the product of an integer multiplied by itself. In the case of 400, it can be expressed as 20 × 20, which equals 400. Therefore, 400 is indeed a perfect square.

## Is the square root of 400 a natural number?

Yes, the square root of 400 is a natural number.

A natural number is a positive integer, and the square root of 400 is exactly 20, which is a positive integer. Therefore, the square root of 400 qualifies as a natural number.

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