113 LearnersLast updated on May 26th, 2025
Square Root of -147
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.
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.
Mistake 1
Ignoring the Imaginary Unit
A common mistake is to ignore the imaginary unit when finding the square root of a negative number. Remember that the square root of a negative number involves the imaginary unit i.
For example, √-50 = 7.07i, not just 7.07.
Mistake 2
Confusing Real and Imaginary Roots
Another mistake is confusing real and imaginary roots. Ensure students understand that negative numbers have imaginary roots, not real ones.
For example, √-147 should be expressed as 12.12i.
Mistake 3
Mistakes in Prime Factorization
When using the prime factorization method, students might incorrectly factorize the number. Ensure the factors are correct, such as 147 = 3 × 7², to avoid errors in simplification.
Mistake 4
Omitting the Negative Sign
Students may forget to include the negative sign when writing the square root of negative numbers. It’s crucial to remember that √-147 = 12.12i, not just 12.12.
Mistake 5
Misapplying the Long Division Method
Errors in long division may arise from skipping steps or incorrect arithmetic. Ensure the division is accurate when finding square roots of positive numbers, and remember to apply the imaginary unit for negative ones.
Square Root of -147 Examples
Problem 1
If the side length of a square is √-98, what is the area of the square in terms of imaginary numbers?
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.
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?
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
Problem 3
Calculate 3 times the square root of -147.
36.372i
Explanation
Find the square root of -147, which is 12.124i.
Then multiply by 3: 3 × 12.124i = 36.372i
Problem 4
What is the result of adding √-147 and √-3?
The result is 14.045i.
Explanation
Find the square roots: √-147 = 12.124i √-3 = 1.732i
Add them: 12.124i + 1.732i = 14.045i
Problem 5
If the hypotenuse of a right triangle is √-147, what is the length in terms of imaginary numbers?
The hypotenuse length is 12.124i units.
Explanation
The hypotenuse is given as √-147, which equals 12.124i units.
FAQ on Square Root of -147
1.What is √-147 in its simplest form?
2.What is i in the context of square roots?
3.Why does -147 not have a real square root?
4.How do you express the square root of a negative number?
5.What is the approximate value of √147?
6.How does learning Algebra help students in Vietnam make better decisions in daily life?
7.How can cultural or local activities in Vietnam support learning Algebra topics such as Square Root of -147?
8.How do technology and digital tools in Vietnam support learning Algebra and Square Root of -147?
9.Does learning Algebra support future career opportunities for students in Vietnam?
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.
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