What is Square Root of 10? How to find the value of √10?

Proof that square root of 2 is irrational | Algebra I | Khan Academy
Proof that square root of 2 is irrational | Algebra I | Khan Academy

The square root of number 10 is a number which when multiplied by itself results in 10. So, square root of number 10 is a solution of the equation \(x^2=10\).

Finding the square and square root of a number are inverse operations. To find the square of a number you need to multiply the number by itself.

For example: To find the square of a number p, we will multiply the number by itself, that is,

\(p~\times~p\) = \(p^2\)

However, to find the square root of 10, we will do the inverse operation,

A square root is represented by this radical symbol: ‘ √ ‘.

So, the square root of 10 is expressed as \(\sqrt {10}\) and in exponential form as \(10^{\frac{1}{2}}\).

The number 10 is not a perfect square. Therefore, \(\sqrt {10}\) will result in an irrational number.

So, the value of \(\sqrt {10}\) can be approximated to the nearest integer or to the nearest tenth, hundredth and so on.

Let’s find the square root of 10 using the prime factorization method.

Step 1: Find the prime factors of 10

∴ 10 = 2 × 5

Step 2: Make pairs of identical numbers.

Since we do not have any identical numbers in the prime factorisation we will need to write 2 as \(\sqrt 2\) x \(\sqrt 2\) and 5 as \(\sqrt 5\) x \(\sqrt 5\)

Therefore,

\(10=(\sqrt 2~\times~\sqrt 2~\times~\sqrt 5~\times~\sqrt 5)\)

Step 3: Take a number from each pair and multiply them to get the square root, that is, \(\sqrt 2~\times~\sqrt 5\) .

So,

\(\sqrt {10}=\sqrt 2~\times~\sqrt 5\)

\(\sqrt {10}\) = 1.414 x 2.2360 Substitute\(\sqrt 2\approx 1.414\) and \(\sqrt 5\approx 2.2360\)

\(\sqrt {10}\approx\) 3.162 Multiply

Therefore, the square root of 10 is approximately equal to 3.162

Each positive number has two square roots, one is positive and other is negative as their absolute value is the same.

Let us understand this with example,

3.162 × 3.162 \(\approx\) 10

-3.162 × -3.162 \(\approx\) 10

[Since, multiplying two negatives results in a positive)]

From the above we can say that square of -3.162 is 10 and square of 3.162 is also 10, so we can write-

\(\sqrt{10}\) = ± 3.162

Example 1: Evaluate the expression \((4\sqrt {10}~\times~2\sqrt {10})~-~10\)

Solution:

Given expression: \((4\sqrt {10}~\times~2\sqrt {10})~-~10\)

\(4\sqrt {10}~\times~2\sqrt {10}~-~10 = (8~\times~10)~-~10\) [Simplify]

= 80 – 10 [Multiply]

= 70 [Subtract]

Hence the value of given expression \((4\sqrt {10}~\times~2\sqrt {10})~-~10\) is 70.

Example 2: Joe, a salesman, wants to build a circular floor garage to store goods. He wants the floor to be of radius 10 feet. If he wants to paint the floor for $\(\sqrt{10}\) per square feet, how much money would he have to pay?

Solution:

Given, radius of the circular floor = 10 feet and

Cost of painting the floor = $ \(\sqrt {10}\) per square feet

Total cost = Area of floor in sq. feet x Cost of painting the floor per sq. feet

Area of circular floor = \(\pi r^2\) [Area of circle formula]

\(=3.14~\times~(10)^2\) [Substitute \(\pi\) = 3.14 and r = 10]

= 3.14 x 100 [Square of 10 is 100]

= 314 square feet [Multiply]

Total cost = 314 x \(\sqrt{10}\)

= 314 x 3.162 [Substitute 3.162 for \(\sqrt{10}\) ]

= 992.87 [Multiply]

Hence, Joe has to pay $992.87.

Example 3: Find the value of square root of 1000.

Solution:

We have to find square root of 1000, that is, \(\sqrt{1000}\).

\(\sqrt{1000}=\sqrt{10~\times~100}\) [Write 1000 as 10 x 100]

= \(\sqrt{10~\times~(2~\times~2)~\times~(5~\times~5)}\) [Prime factorize 100]

= \(\sqrt{10}~\times~2~\times~5\) [Simplify]

= 3.162 x 10 [Substitute 3.162 for \(\sqrt{10}\)]

= 31.62 [Multiply]

Hence, the value of the square root of 1000 is 31.62.

10 has two square roots just like any other number.

No, 10 is not a perfect square and hence, \(\sqrt{10}\) is an irrational number.

Prime factorisation and long division methods can be used to find out the square root of a number.

A number that can be expressed as \(\frac{a}{b}\) where a and b are integers and b ≠ 0 is known as a rational number.

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