854 LearnersLast updated on May 26th, 2025
Square Root of 101
The square root of 101 is the value that, when multiplied by itself, gives the original number 101. The number 101 has a unique non-negative square root, called the principal square root.
What Is the Square Root of 101?
The square root of 101 is ±10.0498756211. Basically, finding the square root is just the inverse of squaring a number and hence, squaring 10.0498756211 will result in 101. The square root of 101 is written as √101 in radical form. In exponential form, it is written as (101)1/2
Finding the Square Root of 101
We can find the square root of 11 through various methods. They are:
- Prime factorization method
- Long division method
- Approximation/Estimation method
Square Root of 101 By Prime Factorization Method
The prime factorization of 101 is done by dividing 101 by prime numbers and continuing to divide the quotients until they can’t be divided anymore.
- Find the prime factors of 101
- After factorizing 101, 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 101 = 101 × 1
But here in case of 101, no pairs of factors can be obtained and a single 101 is remaining
So, it can be expressed as √101
√101 is the simplest radical form of √101
Square Root of 101 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 101:
Step 1 : Write the number 101, 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 1. Here, it is1, Because 12=1 < 1.
Step 3 : Now divide 1 by 1 such that we get 1 as quotient and then multiply the divisor with the quotient, we get 1
Step 4: Subtract 1 from 1. Bring down 0 and 1 and place it beside the difference 0.
Step 5: Add 1 to same divisor, 1. We get 2.
Step 6: Now choose a number such that when placed at the end of 2, a 2-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 1. Here, that number is 0.
20×0=0<1.
Step 7: Subtract 1-0=1. Add a decimal point after the new quotient 10, again, bring down two zeroes and make 1 as 100. Simultaneously add the unit’s place digit of 20, i.e., 0 with 20. We get here, 20. Apply Step 5 again and again until you reach 0.
We will show two places of precision here, and so, we are left with the remainder, 17599 (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 10.049….
Square Root of 101 By Approximation
Approximation or estimation of square root is not the exact square root, but it is an estimate.Here, through this method, an approximate value of square root is found by guessing.
Follow the steps below:
Step 1: Find the nearest perfect square number to 101. Here, it is 100 and 121.
Step 2: We know that, √100=10 and √121=11. This implies that √101 lies between 10 and 11.
Step 3: Now we need to check √101 is closer to 10 or 11. Let us consider 10 and 10.5. Since (10)2=100 and (10.5)2=110.25. Thus, √101 lies between 10 and 10.5.
Step 4: Again considering precisely, we see that √101 lies close to (10)2=100. Find squares of (10.02)2=100.4 and (10.1)2= 112.11.
We can iterate the process and check between the squares of 10.03 and 10.05 and so on.
We observe that √101=10.049…
Common Mistakes and How to Avoid Them in the Square Root of 101
When we find the square root of 101, 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
Misunderstanding symbol
Often when √101 is mistaken as 1012 , we square the number 101 and the get the result of 10201. So, understanding of symbol should be clear
Mistake 2
Misunderstanding the concept of irrational numbers.
√101=10.049… which is an irrational number. Hence, it cannot be expressed as a fraction
Mistake 3
Rounding too early
Rounding errors can occur if rounded too early, which can lead to less accuracy
Mistake 4
Incorrect formula for finding square roots through different methods
The Formula should be applied accurately for iterative methods like Long Division method, Prime Factorization method or Estimation method
Mistake 5
Incorrectly applying the square root property
Square root do not distribute over additions. For example, √101 ≠ √100 + √1
Square Root of 101 Examples
Problem 1
If x= √101, what is x²-1 ?
x= √101
⇒ x2 = 101
⇒ x2-1 = 101-1
⇒ x2-1 = 100
Answer : 100
Explanation
we did the square of the given value of x and then subtracted 1 from it.
Problem 2
Simplify 15√101 (15√101+15√101)?
15√101 (15√101+15√101)
= 15√101(15√101×2)
= (15√101)2×2
= 225×101×2
= 45450
Answer : 45450
Explanation
(√101)2= 101, so multiplying the value with 15 in each part and then simplifying
Problem 3
Simplify (√101 + √101) × √101
(√101 + √101) × √101
= (√101)2 + (√101)2
= 101 + 101
= 202
Answer: 202
Explanation
We multiplied the √125 with each of the √125 inside the bracket and solved.
Problem 4
If a=√101, find a²×a
firstly, a=√101
Now, squaring a get,
a2= 101
or, a2=101
a2× a = 101 × √101 = 101√101
Answer : 101√101
Explanation
squaring “a” which is same as squaring the value of √101 resulted to 101. Then solved applying the concept of square root.
Problem 5
Calculate (√101/3 + √101/4)
√101/3 + √101/4
= √101(1/3+ 1/4)
= (7√101)/12
Answer : (7√101)/12
Explanation
From the given expression, we solved by simple divisions and then simple addition.
FAQs on 101 Square Root
1.What is the square of 101?
2.Is the square root of 101 a prime number?
3.Is 101 a perfect square or non-perfect square?
4.Is the square root of 101 a rational or irrational number?
5.What are the factors of 101?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 101?
8.How do technology and digital tools in India support learning Algebra and Square Root of 101?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for Square Root of 101
- 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, 24 = 16, where 2 is the base, 4 is the exponent
- Prime Factorization: Expressing the given expression as a product of its factors. Ex: 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
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