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Square Root of 120: Step-by-Step Solutions

Square Root of 120: Step-by-Step Solutions

Q: 1. What is the value of the square root of 120?

The square root of 120, denoted as √120, is approximately 10.954. Since 120 is not a perfect square, its square root is an irrational number, meaning it has a non-terminating and non-repeating decimal expansion.

Q: 3. Is 120 a perfect square? Explain the reasoning.

No, 120 is not a perfect square. There are two primary reasons based on the properties of square numbers:Unit Digit Rule: Perfect squares can only end in the digits 0, 1, 4, 5, 6, or 9. Since 120 ends in 0, it could potentially be a perfect square, but a number ending in a single zero is never a perfect square (it must end in an even number of zeroes).Prime Factorization Rule: For a number to be a perfect square, all its prime factors must occur in pairs. The prime factorization of 120 is 2³ × 3¹ × 5¹, where the factors 2, 3, and 5 do not have even powers.

Q: 5. Why is the square root of 120 considered an irrational number?

The square root of 120 is an irrational number because 120 is not a perfect square. A rational number can be expressed as a simple fraction (p/q), where p and q are integers. The square root of any integer that is not a perfect square cannot be represented this way. Its decimal representation is non-terminating and non-repeating (10.95445115...), which is the defining characteristic of an irrational number.

Q: 8. Can the repeated subtraction method be used for finding the exact square root of 120?

No, the repeated subtraction method cannot be used to find an exact value for the square root of 120. This method, which involves subtracting successive odd numbers (1, 3, 5, 7,...), only works for perfect squares. When applied to a perfect square, the process will end exactly at zero. For a non-perfect square like 120, you would subtract odd numbers and eventually get a result less than the next odd number to be subtracted, but not zero. This indicates that the original number is not a perfect square.

Reviewed by:

Rama Sharma

How to Calculate the Square Root of 120 Using Different Methods

Every non-negative real number has a unique non-negative square root, called the principal square root. Which is denoted by √x, where √ is called the radical sign or radix and the number under the radical sign here x is called the radicand.For example, the principal square root of 9 is 3, denoted by √9 = 3, because 32 = 3 x 3 = 9 and 3 is non-negative. The square root of 120 is that number which when multiplied by itself gives the number 120.

Number 120 is not a perfect square hence the square root of 120 results in a decimal number not the whole number.

Square Root of 120 (\[\sqrt{120}\]) = 10.954

[Image Will be Uploaded Soon]

Square Root of 120

The square root of the rational number, 120 gives a decimal number and not a whole number, because the number 120 doesn’t have proper factors. Every non-negative real number has a non-negative square root, which is called the principal square root. Therefore, the square root of 120 gives a result as a unique non-negative number. It is denoted as √120, where √ is called radical sign or radix and the number 120, whose square root has to be derived is called radicand

How to Calculate Square Root of 120

Calculating the square root of an imperfect square is a bit of a complex task. Following are the methods that can be used to find the square root of a number.

Prime factorization method
Repeated subtraction method
Long division method
Number line method
Average method

This method works if the number is a perfect square, but, if the number is not a perfect square prime factorization method and the repeated subtraction method will not work, we have to use other methods for finding the square roots.

Finding the Square Root of 120 by Average Method

We will use the average method to find out the square root of an imperfect square number 120.

Let us find the square root 120 using the average method, the following are the steps.

Step 1: Find out the two perfect square numbers which are very close to the given number on either side. For example the number ‘120’, the immediate perfect square lesser than 120 is ‘100’ and the immediate perfect square greater than 120 is ‘121’.

Step 2 : Note down the square roots of the perfect squares, here the square root of ‘100’ is ‘10’ and the square root of ‘121’ is ‘11’.

Step 3 : Square root of a given number lies between the square roots of numbers determined in step 2.Square root of ‘120’ is any number between 10 and 11.

Step 4: Divide the number whose square root is determined by any of the numbers obtained in Step 2.‘120’ can be divided either by ‘10’ or ‘11’.

Let us divide ‘120’ by ‘10’

\[\frac{120}{10}\] = 12

Step 5: Find the average of the quotient and divisor in Step 4.

The average of 10 and 12 is

Average = \[\frac{10 + 12}{2}\] = 11

Step 6: Now divide 120 by step 5 answer

\[\frac{120}{11}\] = 10.9090

Step 7: Now, average 11 and 10.9090 by adding them together and dividing the sum by two you get 10.954545

Check your work by multiplying your answer by itself. If 10.9545 is multiplied by 10.9545 we get 120.001.

Therefore \[\sqrt{120}\] = 10.9545

Finding Square Root of 120 by Factors

One of the methods to find the square root of 120 is by finding the factors of 120.

Factors of 120 are 2 x 2 x 2 x 3 x 5.

Factor tree for 120 is

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Steps to find out the square root by factors

Step 1: find out the factors of the number here

120 = 2 x 2 x 2 x 3 x 5

Step2 : Now taking root of the factors.

\[\sqrt{120}\] = \[\sqrt{2 \times 2 \times 2 \times 3\times 5}\]

Step 3: taking out root means taking out the pair of common factors as a single common number outside the radical sign. Here only 2 are in pairs so taking out 2 out of the radical sign. And single 2 3 and 5 does not have pair so they remain inside the radical sign

= 2 x \[\sqrt{2 \times 3 \times 5}\]

Step 4: Multiplying, we get

= 2 x\[\sqrt{30}\]

Step 5: finding the root value

= 2 x 5.477

Step 6 : simplify

=10.9544

Finding Square root of 120 by long Division Method

The most precise and convenient method to find out the square root of imperfect square is by long division method. Here is the stepwise calculation by long division method.

10.954

1 20

-1

0 20

209

+ 9

2000

- 1881

2185

11900

- 10925

21904

+ 4

97500

- 87616

21908

9884

Fun Facts

Technically, just the "check mark" part is used in the radical; the line across the top is called the "vinculum".
Square Root Day- The 4th of April 2016 is a Square Root Day, because the date looks like a square root 4/4/16
The next after that is the 5th of May 2025 (5/5/25)

\[\sqrt[Happy]{\text{Square root Day}}\]

Solved Example

Example 1: Find the square root of 12

Solution:

Factor of 12 are 2 x 2 x 3

Taking root

= \[\sqrt{2 \times 3 \times 3}\]

= \[2\sqrt{3}\]

= 2 x 1.7320

=3.4641

Example 2 :Find the square root of 75

Solution:

Factor of 75 = 5 x 5 x 3

Taking the root of

= \[\sqrt{5 \times 5 \times 3}\]

= \[5\sqrt{3}\]

= 5 x 1.7320

= 8.66

Quiz Time:

Find the square root of 86 by long division method.
Find the square root of 45 by factor method.

FAQs on Square Root of 120: Step-by-Step Solutions

1. What is the value of the square root of 120?

The square root of 120, denoted as √120, is approximately 10.954. Since 120 is not a perfect square, its square root is an irrational number, meaning it has a non-terminating and non-repeating decimal expansion.

2. How do you express the square root of 120 in its simplest radical form?

To simplify √120, we use the prime factorization method. The prime factors of 120 are 2 × 2 × 2 × 3 × 5. We look for pairs of identical factors.

Step 1: Write the factorization under the radical: √(2 × 2 × 2 × 3 × 5).
Step 2: Group the pair of 2s: √((2 × 2) × 30).
Step 3: For each pair, take one factor outside the radical: 2√30.

Therefore, the simplest radical form of the square root of 120 is 2√30.

3. Is 120 a perfect square? Explain the reasoning.

No, 120 is not a perfect square. There are two primary reasons based on the properties of square numbers:

Unit Digit Rule: Perfect squares can only end in the digits 0, 1, 4, 5, 6, or 9. Since 120 ends in 0, it could potentially be a perfect square, but a number ending in a single zero is never a perfect square (it must end in an even number of zeroes).
Prime Factorization Rule: For a number to be a perfect square, all its prime factors must occur in pairs. The prime factorization of 120 is 2³ × 3¹ × 5¹, where the factors 2, 3, and 5 do not have even powers.

4. What are the steps to find the value of √120 using the long division method?

The long division method provides an approximate decimal value for the square root of non-perfect squares like 120. Here are the steps:

Step 1: Pair the digits of 120 from the right, so we have 1 and 20.
Step 2: Find the largest number whose square is less than or equal to the first pair (1). This is 1 (since 1² = 1). Write 1 as the quotient and divisor. Subtract 1 from 1, leaving 0.
Step 3: Bring down the next pair (20). The new dividend is 20.
Step 4: Double the quotient (1), which gives 2. Place a blank next to it, making the new divisor 2_.
Step 5: Find a digit for the blank such that 2_ × _ is less than or equal to 20. This digit is 0 (20 × 0 = 0). The quotient is now 10.
Step 6: Place a decimal point in the quotient and bring down a pair of zeros (00). The new dividend is 2000. Double the quotient (10) to get 20. The new divisor is 20_.
Step 7: Find a digit for the blank such that 20_ × _ ≤ 2000. This digit is 9 (209 × 9 = 1881). The quotient is now 10.9.

Continuing this process gives the value 10.954...

5. Why is the square root of 120 considered an irrational number?

The square root of 120 is an irrational number because 120 is not a perfect square. A rational number can be expressed as a simple fraction (p/q), where p and q are integers. The square root of any integer that is not a perfect square cannot be represented this way. Its decimal representation is non-terminating and non-repeating (10.95445115...), which is the defining characteristic of an irrational number.

6. Between which two consecutive whole numbers does the square root of 120 lie?

To estimate the location of √120, we identify the perfect squares closest to 120. We know that:

10² = 100
11² = 121

Since 120 lies between 100 and 121, its square root must lie between the square roots of these two numbers. Therefore, the square root of 120 lies between 10 and 11.

7. What is the key difference between the 'square of 120' and the 'square root of 120'?

The 'square of 120' and the 'square root of 120' are inverse mathematical operations with fundamentally different meanings:

The Square of 120 (120²) means multiplying 120 by itself: 120 × 120 = 14,400.
The Square Root of 120 (√120) is the number that, when multiplied by itself, equals 120. This value is approximately 10.954.

In essence, squaring a number makes it much larger, while finding the square root of a number (greater than 1) makes it smaller.

8. Can the repeated subtraction method be used for finding the exact square root of 120?

No, the repeated subtraction method cannot be used to find an exact value for the square root of 120. This method, which involves subtracting successive odd numbers (1, 3, 5, 7,...), only works for perfect squares. When applied to a perfect square, the process will end exactly at zero. For a non-perfect square like 120, you would subtract odd numbers and eventually get a result less than the next odd number to be subtracted, but not zero. This indicates that the original number is not a perfect square.

9. How might the concept of the square root of 120 be applied in a real-world scenario?

A common real-world application of square roots is in geometry, specifically with area. For example, if an architect is designing a square-shaped room or a plaza that must have an area of 120 square feet, they would need to calculate the length of each side. The side length would be the square root of the area (√120). Calculating this value, approximately 10.954 feet, would be crucial for creating blueprints and ordering materials.