A composite number has more two factors. As we know, 117 is a composite number. When we take square root of 117, we can see that **√**117 can be simplified as 3 **√**13. Thus, **√**117 is an irrational number. In this mini-lesson we will learn to find the square root of 117 by long division method. We will also solve some examples to understand this concept better. Let us now find the square root of 117.

**Square Root of 117**:**√**117 = 10.816**Square of 117: 117**= 13,689^{2}

**Table of Contents**

- What Is the Square Root of 117?
- Is Square Root of 117 Rational or Irrational?
- How to Find the Square Root of 117?
- Important Notes
- FAQs on Square Root of 117
- Thinking Out of the Box!
- Solved Examples on Square Root of 117
- Interactive Questions on Square Root of 117

## What Is the Square Root of 117?

The square root of 117 can be obtained by finding the number whose square gives 117. There are no integers whose square will give the value 117. Hence, **√**117 is given as,

**√**117 = 10.816

On squaring 10.816, we get (10.816)^{2} = 116.985856.. This number is close to 117 when it is rounded to its nearest value.

## Is the Square Root of 117 Rational or Irrational?

As **√**117 = 10.81665382639197.... We can see this is non-terminating and the decimal part has no repeating pattern. A rational number is either terminating or non-terminating and has a repeating pattern in its decimal part. Hence, this shows **√**117 is an irrational number.

## How to Find the Square Root of 117?

There are different methods to find the square root of any number. Click here to know more about the different methods.

### Simplified Radical Form of Square Root of 117

As 117 is a composite number, it can be factored as 117 = (3 × 3 × 13). Hence, square root of 117 can be expressed as **√**(3 × 3 × 13). Thus, the simplified radical form of **√**117 is 3 **√**13.

### Square Root of 117 by Long Division Method

The square root of 117 can be found using the long division method as follows.

**Step 1**: First, we will pair the digits of 117 from right to left. Place a horizontal bar to indicate the pairs. 17 is one such pair, which leaves 1 as a single digit in the other pair.**Step 2**:**Step 3**:**Step 4**: 0 is placed at the units place of the new divisor because when 20 is multiplied by 0, we get 0. The answer obtained now is 17 and we bring down 00.**Step 5**: The quotient now becomes 10 and it is multiplied by 2. This gives 20, which becomes the starting digits of the new divisor.**Step 6**: 8 is placed at the units place of the new divisor because on multiplying 208 by 8 we get 1664. The remainder obtained is 36 and we bring 00 down.**Step 7**: The quotient 108 when multiplied by 2 gives 216, which will be the starting digits of the new divisor.**Step 8**: 1 is placed at the units place of the divisor because on multiplying 2161 by 1, we will get 2161. The answer obtained is 1439. Next, we will bring down the next pair of numbers, 00.**Step 9**: The quotient 1081 when multiplied by 2 gives 2162, which will be the starting digits of the new divisor.**Step 10**: 6 is placed at the units place of the divisor because on multiplying 21626 by 6, we will get 129756. The remainder obtained is 14144. We can bring 00 down and repeat these steps until we get the number we want.

So far we have obtained the value of **√**117 = 10.816. On repeating this process further, we get, **√**117 = 10.81665382639197...

**Explore square roots using illustrations and interactive examples**

**Important Notes:**

- The two perfect squares closest to 117 are 100 and 121. On taking the square root on both the sides, we can see that
**√**117 lies between**√**100 and**√**121 i.e.,**√**117 lies between 10 and 11. - 117 is not a perfect square. The square root of 117 is irrational.

## FAQs on Square Root of 117

### Can the square root of 117 be simplified?

The square root of 117 can be simplified as 3**√**13.

### What is the square root of 117 rounded to its nearest tenth?

We know that, **√**117 = 10.816. On rounding the square root of 117 to its nearest tenth, we get 10.8.

### Does 117 have a square root?

Yes, 117 has a square root but it is not a whole number as it is not a perfect square number.

### What is the value of the square root of 117?

The value of the square root of 117 is **√**117 = 10.81665382639197(approx.).

### Is the square root of 117 rational or irrational?

The square root of 117 is irrational.

### Is the square root of 117 a real number?

Yes, the square root of 117 is a real number.

**Think Tank:**

- Can you find the value of
**√**(-117)? - Can you determine if the value of
**√**(-117) is the same as -**√**117?

## Square Root of 117 Solved Examples

**Example 1**: Determine whether the radius of a circle having an area 117π square inches is less than 12 inches or not.

**Solution**

Let us assume that the radius of the circle is x inches. Then the area of the circle is πx^{2}. From the given information, we have:

x^{2 }= 117

x = **√**117 = 10.8 inches (approx.)

The radius of the circle is 10.8 inches. Hence, the radius of the circle having an area 117π square inches is less than 12 inches.

**Example 2**: What is the length of a square room with area 117 square feet?

**Solution**

For finding the length of the room, we use the formula, area = (length)^{2} = 117 square feet

length^{2} = 117

length = **√**117

On calculating the square root of 117, we get the result as 10.8 feet. Therefore, the length of the square room with area 117 square feet is 10.8 feet.

## Interactive Questions

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**