Root 10- Value, derivation and examples

The value of root 10 

Root 10 is denoted as √10 in its radical form. Its square root (positive), rounded up to three decimal spaces, is 3.162. 

What are square roots 

The square root of a number is the value which, when multiplied by itself gives the original value. It is the inverse of squares.

About root 10 

The value of root 10 is an irrational number. The decimal form of √10 is non terminating. At the same time, it is non-recurring. That means none of its digits in the decimal space ever repeat themselves nor can we see a pattern in their appearance.

Here, we’re going to have a look at the methods of finding out the value of √10.

Determining the value of √10 using the long division method

The long division method is one of the most trusted ways of determining the root values of numbers. Here, we’ll follow these steps to find the square root of 10 by long division method:-

Step 1:

Make a pair of digits (by placing a bar over it) from the unit’s place and place this inside the radical symbol.

Step 2:

We’ll now have to find a number such that when it is multiplied with itself, the product is less than or equal to 10

We know that 3² gives us 9 and 9 is less than 10. So it becomes our divisor.

Step 3:

Next, we’ll place a decimal point and a pair of zeros next to 10 (dividend) and continue our division. Now, we multiply the quotient by 2 and the product becomes the starting digit of our next divisor.

Step 4:

Now we’ll have to choose a number in the unit’s place for the new divisor such that its product with a number is less than or equal to 100.

So, the closest multiplication we’re left with is 61 × 1 = 61

Step 5:

We bring down the next pair of zeros and multiply the quotient by 2, and they become the two starting digits of the new divisor.

Step 6:

Now, we choose a number in the unit’s place for the new divisor such that its product with a number is less than or equal to 3900. 3756 is the nearest number possible, leaving us with a remainder of 144.

Step 7:

More pairs of zeros are added and the process is repeated to find the new divisor and product as in step 2. Root 10 being irrational this division process is non terminating and will go on till ‘n’ number of terms.

Illustrations

1. Solve for x : 4x² = 2x² +20

4x²- 2x² = 20

2x² = 20

x² = 10 x= √10

Therefore, x = ±3.162 (truncated)

2. Find the value of √10

√10 can be simplified as = √(2×5)

We know, √2 = 1.414 and  √5 = 2.23

So, 

√10 = 1.414 x 2.23

Therefore, 

the value of √10 = 3.162