Square Root of -147
2026-02-28 08:40 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-147

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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of -147.

What is the Square Root of -147?

The square root is the inverse of the square of the number. Since -147 is a negative number, it does not have a real square root because a real number squared is always non-negative. Instead, the square root of -147 is expressed in terms of an imaginary number. In radical form, it is expressed as √-147, which can be written as i√147 in terms of real and imaginary components, where i is the imaginary unit (i² = -1). The approximate value of √147 is 12.124, so the square root of -147 is approximately 12.124i.

Finding the Square Root of -147

To find the square root of a negative number, we use imaginary numbers. Here, we will illustrate how to express the square root of -147 using imaginary numbers:

Step 1: Express the negative number in terms of a positive number and the imaginary unit: √-147 = √(147) × √(-1) = √147 × i

Step 2: Calculate the square root of 147, which is approximately 12.124.

Step 3: Combine the result with the imaginary unit: √-147 = 12.124i

Square Root of 147 by Prime Factorization Method

To find the square root of 147, we can use the prime factorization method for the positive part of the number:

Step 1: The prime factorization of 147 is 3 × 7 × 7 or 3 × 7².

Step 2: Pair the prime factors: Since we have a pair of 7s, we can take one 7 out of the square root.

Step 3: The square root of 147 in simplest form is expressed as √147 = √(3 × 7²) = 7√3.

Thus, the square root of -147 is 7√3i.

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Square Root of -147 by Long Division Method

The long division method is typically used for finding square roots of non-perfect squares. For -147, we are interested in the imaginary square root:

Step 1: Consider the positive part, 147, and find its square root using long division, which is approximately 12.124.

Step 2: Since -147 is negative, the square root is expressed as an imaginary number: √-147 = 12.124i.

Square Root of -147 by Approximation Method

To approximate the square root of -147, we use the square root of 147 and express it in terms of an imaginary number:

Step 1: Approximate √147 using the closest perfect squares, 144 and 169. √147 lies between 12 and 13.

Step 2: Calculate the decimal approximation: (147 - 144) / (169 - 144) = 3 / 25 = 0.12

Step 3: Add the decimal approximation to the lower bound: 12 + 0.12 = 12.12

Step 4: Therefore, the square root of -147 is approximately 12.12i.

Common Mistakes and How to Avoid Them in the Square Root of -147

Students often make mistakes when dealing with negative square roots, such as ignoring the imaginary unit or incorrect calculations. Let’s explore some common errors and how to prevent them.

Problem 1

If the side length of a square is √-98, what is the area of the square in terms of imaginary numbers?

Okay, lets begin

The area of the square is -98 square units.

Explanation

The area of the square = side².

The side length is given as √-98.

Area = (√-98)² = -98

Therefore, the area of the square is -98 square units.

Well explained 👍

Problem 2

A rectangular garden has a length of √-147 meters and a width of 10 meters. What is the perimeter in terms of imaginary numbers?

Okay, lets begin

The perimeter is 20 + 24.248i meters.

Explanation

Perimeter of a rectangle = 2 × (length + width)

Perimeter = 2 × (√-147 + 10) = 2 × (12.124i + 10) = 20 + 24.248i meters

Well explained 👍

Problem 3

Calculate 3 times the square root of -147.

Okay, lets begin

36.372i

Explanation

Find the square root of -147, which is 12.124i.

Then multiply by 3: 3 × 12.124i = 36.372i

Well explained 👍

Problem 4

What is the result of adding √-147 and √-3?

Okay, lets begin

The result is 14.045i.

Explanation

Find the square roots: √-147 = 12.124i √-3 = 1.732i

Add them: 12.124i + 1.732i = 14.045i

Well explained 👍

Problem 5

If the hypotenuse of a right triangle is √-147, what is the length in terms of imaginary numbers?

Okay, lets begin

The hypotenuse length is 12.124i units.

Explanation

The hypotenuse is given as √-147, which equals 12.124i units.

Well explained 👍

FAQ on Square Root of -147

1.What is √-147 in its simplest form?

The simplest form of √-147 is 7√3i, derived from the prime factorization of 147 as 3 × 7².

2.What is i in the context of square roots?

The imaginary unit i is defined as √-1, used to represent the square root of negative numbers.

3.Why does -147 not have a real square root?

Negative numbers do not have real square roots because a real number squared is always non-negative. The square root of a negative number is expressed using the imaginary unit i.

4.How do you express the square root of a negative number?

The square root of a negative number is expressed using the imaginary unit i. For example, √-147 = √147 × i.

5.What is the approximate value of √147?

The approximate value of √147 is 12.124, used to express the square root of -147 as 12.124i.

Important Glossaries for the Square Root of -147

  • Imaginary Unit: Represented by i, it is the square root of -1, used to express square roots of negative numbers.
  • Square Root: The value that, when multiplied by itself, gives the original number. For negative numbers, it includes the imaginary unit.
  • Prime Factorization: The expression of a number as the product of its prime factors, used for simplifying square roots.
  • Approximation: A method of finding a near value, applied to non-perfect square roots for estimating decimal values.
  • Real Number: A value that represents a quantity along a continuous line, excluding imaginary numbers like those involving i.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.