366 LearnersLast updated on May 26th, 2025
Square Root of 125
The square root of 125 is a value “y” such that when “y” is multiplied by itself → y × y, the result is 125. The number 125 has a unique non-negative square root, called the principal square root.
What Is the Square Root of 125?
The square root of 125 is ±11.1803398875. Finding the square root is just the inverse of squaring a number and hence, squaring 11.1803398875 will result in 125. The square root of 125 is written as √125 in radical form. In exponential form, it is written as (125)1/2
Finding the Square Root of 125
We can find the square root of 125 through various methods. They are:
- Prime factorization method
- Long division method
- Approximation/Estimation method
Square Root of 125 By Prime Factorization Method
The prime factorization of 125 is done by dividing 125 by prime numbers and continuing to divide the quotients until they can’t be divided anymore.
Find the prime factors of 125
After factorizing 125, 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 125 = 5 × 5 × 5
But here in case of 125, a pair of factors 5 can be obtained and a single 5 is remaining
So, it can be expressed as √125 = 5 × √5 = 5√5
5√5 is the simplest radical form of √125
Square Root of 125 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 125:
Step 1 : Write the number 125, 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 is 1, 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 2 and 5 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 25. Here, that number is 1.
21×1=21<25.
Step 7: Subtract 25-21=4. Add a decimal point after the new quotient 11, again, bring down two zeroes and make 4 as 400. Simultaneously add the unit’s place digit of 21, i.e., 1 with 21. We get here, 22. 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, 7600 (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 11.180….
Square Root of 125 By Approximation
Follow the steps below:
Step 1: Find the nearest perfect square number to 125. Here, it is 121 and 144.
Step 2: We know that, √121=11 and √144=12. This implies that √125 lies between 11 and 12.
Step 3: Now we need to check √125 is closer to 11 or 12. Let us consider 11 and 11.5. Since (11)2=121 and (11.5)2=132.25. Thus, √125 lies between 11 and 11.5.
Step 4: Again considering precisely, we see that √125 lies close to (11)2=121. Find squares of (11.1)2=123.21 and (11.3)2= 127.69.
We can iterate the process and check between the squares of 11.15 and 11.2 and so on.
We observe that √125=11.180…
Common Mistakes and How to Avoid Them in the Square Root of 125
When we find the square root of 125, 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 √125 is mistaken as 1252 , we square the number 125 and the get the result as 15625. So, understanding of symbol should be clear
Square Root of 125 Examples
Problem 1
What is √125 in the form of fraction?
√125 = 5√5
Since, 5√5 is irrational, √125 cannot be expressed exactly as a simple Fraction.
Answer: √125 cannot be expressed exactly as a simple Fraction.
Explanation
Since, 5√5 is irrational, √125 cannot be expressed exactly as a simple Fraction.
Problem 2
Find the area of a square whose side length is 11.18 cm ?
Given, the side = 11.18 cm
We know that, area of square = (side of a square)2
Or, area of a square = (11.18)2
Or, area of a square = 124.99 cm2
The area of a square with side 11.18 cm is 124.99 cm2
Answer: 124.99 cm2
Explanation
We know that, (side of a square)2 = area of square. Here, we are given with the side length of the square, so, we can easily find out its square value because its square value is the measure of the area of the square
Problem 3
Simplify (√125 + √125) × √125
(√125 + √125) × √125
= (√125)2 + (√125)2
= 125 + 125
= 250
Answer: 250
Explanation
We multiplied the √125 with each of the √125 inside the bracket and solved.
Problem 4
if x= √125, what is x^2-5, x^2+5 ?
x= √125
⇒ x2 = 125
⇒ x2 -5 = 125-5
⇒ x2 -5 = 120
Similarly, x= √125
⇒ x2 = 125
⇒ x2 +5 = 125+5
⇒ x2 +5 = 130
Answer : 120,130
Explanation
We did the square of the given value of x and then subtracted 5 from it.
We did the square of the given value of x and then added 5 to it.
Problem 5
Calculate (√125/5 + √125/10+ √125/15)
√125/5 + √125/10 + √125/15
= 5√5/5 + 5√5/10 + 5√5/15
= √5 + √5/2 +√5/3
= (6√5+3√5+2√5)/6
= (11√5)/6
Answer : (11√5)/6
Explanation
From the given expression, we first found the simplest radical form of square root of 125 then solved by simple divisions and then simple addition.
FAQs on 125 Square Root
1.Can the square root of 125 be negative?
2.How can we solve the square root of 126?
3.Is 125 a perfect square or non-perfect square?
4.Is the square root of 125 a rational or irrational number?
5.What is the cube root of 125?
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 125?
8.How do technology and digital tools in India support learning Algebra and Square Root of 125?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for Square Root of 125
- 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 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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