705 LearnersLast updated on May 26th, 2025
Square root of 400
The square root of 400 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 400. It contains both positive and a negative root, where the positive root is called the principal square root.
What Is the Square Root of 400?
The square root of 400 is ±20. The positive value, 20 is the solution of the equation x2 = 400. As defined, the square root is just the inverse of squaring a number, so, squaring 20 will result in 400. The square root of 400 is expressed as √400 in radical form, where the ‘√’ sign is called the “radical” sign. In exponential form, it is written as (400)1/2
Finding the Square Root of 400
We can find the square root of 400 through various methods. They are:
- Prime factorization method
- Long division method
- Subtraction method
Square Root of 400 By Prime Factorization Method
The prime factorization of 400 involves breaking down a number into its factors. Divide 400 by prime numbers, and continue to divide the quotients until they can’t be separated anymore.
After factoring 400, make pairs out of the factors to get the square root. If there exists numbers which cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs
So, Prime factorization of 400 = 2 × 2 × 2 × 2 × 5 × 5
But for 400, two pairs of factor 2 and one pair of factors 5 can be obtained.
So, it can be expressed as √400 = √(2 × 2 × 2 × 2 × 5 × 5) = 2 × 2 × 5 = 20
20 is the simplest radical form of √400
Square Root of 400 By Long Division Method
This is a method used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.
Follow the steps to calculate the square root of 400:
Step 1: Write the number 400 and draw a bar above the pair of digits from right to left.
Step 2: Now, find the greatest number whose square is less than or equal to 4. Here, it is 2 because 22=4
Step 3: now divide 4 by 2 (the number we got from Step 2) such that we get 2 as a quotient, and we get a remainder. Double the divisor 2, we get 4, and then the largest possible number A1=0 is chosen such that when 0 is written beside the new divisor 4, a 2-digit number is formed →40, and multiplying 0 with 40 gives 0, which is less than or equal to 0.
Repeat this process until you reach the remainder of 0.
Step 4: The quotient obtained is the square root of 400. In this case, it is 20.
Square Root of 400 By Subtraction Method
We know that the sum of the first n odd numbers is n2. We will use this fact to find square roots through the repeated subtraction method. Furthermore, we just have to subtract consecutive odd numbers from the given number, starting from 1. The square root of the given number will be the count of the number of steps required to obtain 0. Here are the steps:
Step 1: take the number 400 and then subtract the first odd number from it. Here, in this case, it is 400-1=399
Step 2: we have to subtract the next odd number from the obtained number until it comes zero as a result. Now take the obtained number (from Step 1), i.e., 399, and again subtract the next odd number after 1, which is 3, → 399-3=396. Like this, we have to proceed further.
Step 3: Now we have to count the number of subtraction steps it takes to yield 0 finally.
Here, in this case, it takes 20 steps.
So, the square root is equal to the count, i.e., the square root of 400 is ±20.
Common Mistakes and How to Avoid Them in the Square Root of 400
When we find the square root of 400, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
Mistake 1
Incorrectly applying the square root property
Square roots do not distribute over additions. For example, √400 ≠ √36+√364
Square Root of 400 Examples
Problem 1
Find √(400⤬4) ?
√(400⤬4)
= 20 ⤬2
= 40
Answer : 40
Explanation
firstly, we found the values of the square roots of 400 and 4, then multiplied the values.
Problem 2
What is √400 multiplied by 4 ?
√400 ⤬ 4
= 20⤬4
= 80
Answer: 80
Explanation
finding the value of √400 and multiplying by 4.
Problem 3
Find the radius of a circle whose area is 400π cm^2.
Given, the area of the circle = 400π cm2
Now, area = πr2 (r is the radius of the circle)
So, πr2 = 400π cm2
We get, r2 = 400 cm2
r = √400 cm
Putting the value of √400 in the above equation,
We get, r = ±20 cm
Here we will consider the positive value of 20.
Therefore, the radius of the circle is 20 cm.
Answer: 20 cm.
Explanation
We know that, area of a circle = πr2 (r is the radius of the circle). According to this equation, we are getting the value of “r” as 20 cm by finding the value of the square root of 400.
Problem 4
Find the length of a side of a square whose area is 400 cm^2
Given, the area = 400 cm2
We know that, (side of a square)2 = area of square
Or, (side of a square)2 = 400
Or, (side of a square)= √400
Or, the side of a square = ± 20.
But, the length of a square is a positive quantity only, so, the length of the side is 20 cm.
Answer: 20 cm
Explanation
We know that, (side of a square)2 = area of square. Here, we are given with the area of the square, so, we can easily find out its square root because its square root is the measure of the side of the square
Problem 5
Find √400 / √100
√400/√100
= √(400/100)
=√4
= 2
Answer : 2
Explanation
we firstly found out the values of √400 and √100, then divided .
FAQs on 400 Square Root
1.Which two perfect squares, when multiplied together, result into 400?
2.Is √400 a natural number?
3.Is 400 a perfect square or non-perfect square?
4.Is the square root of 400 a rational or irrational number?
5.What is the cube root of 400?
6.How does learning Algebra help students in Canada make better decisions in daily life?
7.How can cultural or local activities in Canada support learning Algebra topics such as Square root of 400?
8.How do technology and digital tools in Canada support learning Algebra and Square root of 400?
9.Does learning Algebra support future career opportunities for students in Canada?
Important Glossaries for Square Root of 400
- Exponential form: An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent.Ex: 3 ⤬ 3 ⤬ 3 ⤬ 3 = 81 or, 34 = 81, where 3 is the base, 4 is the exponent.
- Factorization: Expressing the given expression as a product of its factors Ex: 52=2 ⤬ 2 ⤬ 13
- Prime Numbers : Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....
- Rational numbers and Irrational numbers: The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers.
- perfect and non-perfect square numbers: Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :2, 8, 18
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