How to Calculate and Apply the Square Root of 10 in Maths
Square Root
The square root of any number gives the same number when the number multiplied by itself.
For example - \[\sqrt{p\times p}\] = \[\sqrt {(p)^{2}}\] = p
Square Root Symbol
The symbol used to denote the square root is "√". It is also known as a radical symbol or radix. The number written under the square root symbol is called the radicand. The square value can be represented in the radical form as well as in decimal form. Square root of 10 can also be represented as a radical of 10.
How to Calculate the Value of a Square Root of 10?
Calculating the root 10 value is a bit complex because the number 10 is not a perfect square as its unit digit is 0. Square root of a number can be easily obtained if the number is in a perfect square. The number is considered as a perfect square if it can be denoted as a product of two equal integers.
For example- 5x5= 25, it is representing the square of a number 5. It is considered as a perfect square as it is stated as a product of two similar integers i.e. 5 x 5 = 25, 6 x 6= 36. It is representing the square of 6. It is even considered as a perfect square as it is stated as a product of two similar integers i.e. 6 x 6.
The number is a perfect square if the unit place of a number ends with 1,4,5,6 or 9
The number is not said to be a perfect square if it ends with 2, 3, 7 or 8.
We can calculate the value of a square root or root 10 values through two methods.
First method to calculate the value of a square root or root 10 value is to use the unit digit of the given number,
The second method to calculate the square root value or root 10 values of the given number is by using a long division method.
What is the Square Root of 10?
The square root of 10 or root 10 is represented in the form of √10. As we know 10 is an even number but not a prime number. Prime numbers are considered as those numbers which have only two factors i.e. 1 and the number itself. For example 2 is a prime number as it has only factors 1 and 2 itself. But number 10 is not a prime number because it has multiple factors like.
1x10 = 10
2 x 5= 10
10 x 1 = 10
5 x 2 = 10
To calculate the square root of value 10, write its factors first
10 = 2 x 5
Square root of 10 can be written in the below format
\[\sqrt 10\] = \[\sqrt 2\] X \[\sqrt 5\]
Common square terms out of the root in the above equation cannot be taken out as it has no common square terms.
\[\sqrt 10\] =\[\sqrt 2\] \[\sqrt 5\]
\[\sqrt 10\] or root 10 is in the radical form. If you want to write it in a decimal form, then substitute approximate values of \[\sqrt 2\] and \[\sqrt 5\] which is 1.414 and 2.236 respectively,
8\[\sqrt 10\] =1.414 x 2.236
\[\sqrt 10\] =3.162
Hence, the value of root 10 or root 10 is 3.162
Square root of 10 using Long Division Method
(division method image will be uploaded soon)
Square Root Values
Solved Example
1. Find the value of √80 + 16√5 , if 3√5+√125 = 17.88
Solution: \[\frac{x}{\sqrt{512}}\] = \[\frac{\sqrt{648}}{x}\]
= 3√5 + √125 = 17.88
= 3√5 + (√25 x √5) = 17.88
= 3√5 + 5√5 = 17.88
= 8√5 = 17.88
= √5 = 17.88/8
= √80 + 16√5 = √16 x √5 + 16 √5
= 4√5 + 16√5 = 20√5
= 20 x 17.88/8 = 44.7
2. Simplify: (√7 -1/√7)2
= (√7 -1/√7)2
= (√7 -1/√7)2
= (√7)2 – 2 x √7 x 1√7 + (1/√7)2
= 7-2 +1/7
= 5 + 1/7
= 36/7
Fun Facts
The Yale Babylonian has a tablet from nearly 4000 years ago that states the square root of 2 out to 9 decimal places by making use of a square and two diagonals.
Communities in Ancient India were making use of square roots since 800 BCE.
An Indian Mathematician from the 9th century named Mahavira is the first person to announce that negative square roots do not take place.
Procedure to determine the square root is outlined in the Chinese book, Writings on Reckoning, written in around 200 BCE during the Han Dynasty.
Quiz Time
1. A square garden having the area measurement of 225 square feet. How much fencing will be needed by the gardener to purchase to fix fencing around the garden?
60 ft
112.5 ft
15 ft
56.25 ft
2. What will be the length of one side of a square, if the area of a square is 100 meters?
√10
25
10
50
3. What will be the value of : \[\sqrt 0.001\] + \[\sqrt 0.81\] + \[\sqrt 1.21\] + \[\sqrt 0.0009\]
2.13
2.03
2.11
2.1
FAQs on Square Root of 10 Explained
1. What is the approximate value of the square root of 10?
The square root of 10 is an irrational number, which means its decimal form never ends or repeats. For practical calculations, the value of the square root of 10 (√10) is approximated to 3.162. This value is commonly used in problems where precision up to three decimal places is required.
2. How is the square root of 10 calculated using the long division method?
The long division method is a standard technique taught in the CBSE syllabus to find the value of non-perfect squares like 10. The process is as follows:
- Step 1: Write 10 as 10.0000 and group the digits in pairs. The first group is 10.
- Step 2: Find the largest integer whose square is less than or equal to 10. This is 3, since 3² = 9. This will be the first digit of the answer.
- Step 3: Subtract 9 from 10, leaving a remainder of 1. Bring down the next pair of zeros (00), making the new dividend 100.
- Step 4: Double the current quotient (3) to get 6. Find a digit 'x' such that 6x multiplied by x is close to 100. Here, x = 1 (since 61 × 1 = 61).
- Step 5: The quotient is now 3.1. The process is repeated to find more decimal places.
3. Is the square root of 10 a rational or an irrational number?
The square root of 10 is an irrational number. A number is rational if it can be written as a fraction p/q, where p and q are integers. This is only possible if the number under the square root is a perfect square. Since 10 is not a perfect square (no integer multiplied by itself equals 10), its square root is irrational, and its decimal representation is non-terminating and non-repeating.
4. What is the square root of 10 in its simplest radical form?
To simplify a radical, we look for perfect square factors within the number. The prime factorisation of 10 is 2 × 5. Since there are no pairs of identical prime factors, no number can be moved outside the square root symbol. Therefore, √10 is already in its simplest radical form.
5. Why can't the square root of 10 be expressed as a simple fraction?
This is a fundamental property of irrational numbers. For a square root to be rational (and thus expressible as a simple fraction), the number inside the root must be a perfect square. A perfect square is the result of an integer multiplied by itself (e.g., 9 is a perfect square from 3 × 3). Since there is no integer that, when squared, equals 10, its square root is irrational and cannot be accurately written as a fraction.
6. Between which two consecutive integers does the square root of 10 lie?
To estimate where √10 lies, we can compare 10 to the nearest perfect squares:
- The perfect square just below 10 is 9 (which is 3²).
- The perfect square just above 10 is 16 (which is 4²).
7. If a square has an area of 10 square units, what is the exact length of its side?
The formula for the area of a square is Area = side². To find the length of a side given the area, you must take the square root of the area. For a square with an area of 10 square units, the length of one of its sides is exactly √10 units. This is a common real-world application of square roots in geometry.
8. Why is 10 not considered a perfect square?
A number is a perfect square if its square root is a whole number. We can verify this by looking at its prime factors. The prime factorisation of 10 is 2 × 5. For a number to be a perfect square, all of its prime factors must exist in pairs. In this case, the factors 2 and 5 both appear only once. Because there are no pairs of prime factors, 10 is not a perfect square.
