401 LearnersLast updated on May 26th, 2025
Square Root of 10
The square root of 10 is a value “y” such that when “y” is multiplied by itself → y × y, the result is 10. The number 10 has a unique non-negative square root, called the principal square root.
What is the Square Root of 10?
The square root of 10 is ±3.16227766017. Finding the square root is just the inverse of squaring a number and hence, squaring 3.16227766017 will result in 10. The square root of 10 is written as √10 in radical form. In exponential form, it is written as (10)1/2
Finding the Square Root of 10
We can find the square root of 10 through various methods. They are:
- Prime factorization method
- Long division method
- Approximation/Estimation method
Square Root of 10 By Prime Factorization Method
The prime factorization of 10 is done by dividing 10 by prime numbers and continuing to divide the quotients until they can’t be separated anymore.
Steps for Prime Factorization of 10:
Find the prime factors of 10.
After factorizing 10, make pairs out of the factors to get the square root.
If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.
So, Prime factorization of 10 = 5 × 2
But here in case of 10, no pair of factors can be obtained but a single 2 and a single 5 are remaining.
So, it can be expressed as √10 = √(5 × 2) = √10
√10 is the simplest radical form of √10
Square Root of 10 By Long Division Method
This method is 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 10:
Step 1 : Write the number 10, and draw a horizontal 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 10. Here, it is 3, Because 32=9 < 10.
Step 3 : Now divide 10 by 3 (the number we got from Step 2) such that we get 3 as quotient and then multiply the divisor with the quotient, we get 9.
Step 4: Subtract 9 from 10, we get 1. Add a decimal point after the quotient 3, and bring down two zeroes and place it beside 1 to make it 100.
Step 5: Add 3 to same divisor, 3. We get 6.
Step 6: Now choose a number such that when placed at the end of 6, a 2-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 100. Here, that number is 1.
61×1=61<100.
Step 7: Subtract 100-61=39. Again, bring down two zeroes and make 39 as 3900. Simultaneously add the unit’s place digit of 61, i.e., 1 with 61. We get here, 62. Apply step 5 again and again until you reach 0.
We will show 2 steps of precision here, and so, we are left with the remainder, 1756 (refer to the picture), after some iterations and keeping the division till here, at this point
Step 8: The quotient obtained is the square root. In this case, it is 3.1627….
Square Root of 10 By Estimation method
Estimation refer to a reasonable guess of the actual value to make calculations realistic and precise. This method helps in estimating the square root of a number.
Step 1: Find the nearest perfect square number to 10. Here, it is 9 and 16.
Step 2: We know that, √9=3 and √16=4. This implies that √10 lies between 3 and 4.
Step 3: Now we need to check √10 is closer to 3 or 4. Let us consider 3 and 3.5, since (3)2=9 and (3.5)2=12.25. Thus, √10 lies between 3 and 3.5.
Step 4: Again considering precisely, find squares of (3.1)2=9.61 and (3.3)2= 10.89.
We can iterate the process and check between the squares of 3.15 and 3.2 and so on.
We observe that √10=3.162…
Common Mistakes and How to Avoid Them in the Square Root of 10
When we find the square root of 10, 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
Assuming √10 as a simple fraction
√10=3.162… which is an irrational number. Hence, it cannot be expressed as a fraction
Square Root of 10 Examples
Problem 1
if x= √10, what is x^2-5 ?
x= √10
⇒ x2 = 10
⇒ x2-5 = 10-5
⇒ x2-5
= 5
Answer : 5
Explanation
we did the square of the given value of x and then subtracted 5 from it.
Problem 2
Find the length of a side of a square whose area is 10 cm^2
Given, the area = 10 cm2
We know that, (side of a square)2 = area of square
Or, (side of a square)2 = 10
Or, (side of a square)= √10
Or, the side of a square = ±3.16227
But, length of a square is a positive quantity only, so, length of the side is 3.16227 cm.
Answer: 3.16227 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 3
Simplify (√10 + √10) ⤫ √10
(√10 + √10) ⤫ √10
= (3.16227 + 3.16227) ⤫ 3.16227
= 6.32454 ⤫ 3.16227
= 19.999
Answer: 19.999
Explanation
We first solved the part inside the brackets, i.e., √10 + √10, which resulted into 6.32454 and then multiplying it with √10 which is 3.16227. we get 19.999
Problem 4
If y=√10, find y^2
firstly, y=√10= 3.16227
Now, squaring y, we get,
y2= (3.16227)2=10
or, y2=10
Answer : 10
Explanation
squaring “y” which is same as squaring the value of √10 resulted to 10
Problem 5
Calculate (√10/3 + √10/5)
√10/3 + √10/5
= 3.16227/ 3 + 3.16227/5
= 1.05409 + 0.6324
= 1.686544
Answer : 1.686544
Explanation
From the given expression, we first found the value of square root of 10 then solved by simple divisions and then simple addition.
FAQs on 10 Square Root
1.How to solve √50?
2.Is the square root of 10 a whole number?
3.Is 10 a perfect square or a non-perfect square?
4.Is the square root of 10 a rational or irrational number?
5.How to solve √30?
6.How does learning Algebra help students in Singapore make better decisions in daily life?
7.How can cultural or local activities in Singapore support learning Algebra topics such as Square Root of 10?
8.How do technology and digital tools in Singapore support learning Algebra and Square Root of 10?
9.Does learning Algebra support future career opportunities for students in Singapore?
Important Glossaries for Square Root of 10
- 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: 2 × 2 × 2 × 2 = 16 Or, 2 4 = 16, where 2 is the base, 4 is the exponent
- Prime Factorization: Expressing the given expression as a product of its factorsEx: 48=2 × 2 × 2 × 2 × 3
- 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 :3, 8, 24
- Base number: The number that is being multiplied by itself a particular amount of times.
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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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