Square Root of -175
2026-02-28 10:24 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-175

205 Learners

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 various fields such as engineering and science. Here, we will discuss the square root of -175.

What is the Square Root of -175?

The square root is the inverse of the square of the number. Since -175 is a negative number, its square root is not a real number. Instead, it is expressed in terms of imaginary numbers. The square root of -175 can be expressed in radical form as √(-175) or as 5i√7, where i is the imaginary unit.

Understanding the Square Root of -175

For negative numbers, the square root involves the imaginary unit "i," which is defined as √(-1). The square root of -175 can be found by factoring -175 into -1 and 175. We express this as √(-175) = √(-1×175) = √(-1)×√(175) = i√175. Since 175 = 5×5×7, we can further simplify this to 5i√7.

Square Root of -175 by Prime Factorization Method

The prime factorization of 175 is 5×5×7. To express the square root of -175 using prime factorization:

Step 1: Factor 175 into prime factors: 5×5×7.

Step 2: To find the square root of -175, express it as √(-1×5×5×7).

Step 3: Simplify the square root of the positive part: √(5×5×7) = 5√7.

Step 4: Combine with the imaginary unit: √(-175) = i×5√7 = 5i√7.

Explore Our Programs

Square Root of -175 by Approximation Method

Approximation for imaginary numbers involves estimating the absolute value and then multiplying by i:

Step 1: First, approximate the square root of the absolute value, 175.

Step 2: Since 175 is between 144 (12^2) and 196 (14^2), we approximate √175 ≈ 13.2.

Step 3: Combine with the imaginary unit: The approximate value of √(-175) is 13.2i.

Common Mistakes and How to Avoid Them in the Square Root of -175

Students often make errors when working with the square root of negative numbers, such as ignoring the imaginary unit or incorrectly simplifying expressions. Let’s look at some common mistakes and how to avoid them.

Problem 1

Can you help Max find the magnitude of a vector if its component along one axis is √(-175)?

Okay, lets begin

The magnitude is 5√7.

Explanation

The magnitude of a vector with an imaginary component is the absolute value. If the component is √(-175), the magnitude is |5i√7| = 5√7.

Well explained 👍

Problem 2

Calculate √(-175) × 2.

Okay, lets begin

The result is 10i√7.

Explanation

First, find the square root of -175, which is 5i√7.

Then multiply by 2: 5i√7 × 2 = 10i√7.

Well explained 👍

Problem 3

What is the square of √(-175)?

Okay, lets begin

The square is -175.

Explanation

Squaring the square root returns the original number: (√(-175))^2 = -175.

Well explained 👍

FAQ on Square Root of -175

1.What is √(-175) in terms of i?

The square root of -175 in terms of i is expressed as 5i√7.

2.What are the prime factors of 175?

The prime factors of 175 are 5 and 7 (175 = 5×5×7).

3.Is the square root of -175 a real number?

No, the square root of -175 is not a real number; it is an imaginary number.

4.What is the absolute value of √(-175)?

The absolute value of √(-175) is 5√7.

5.How do you express √(-175) using imaginary numbers?

The square root of -175 is expressed as 5i√7 using imaginary numbers.

Important Glossaries for the Square Root of -175

  • Square root: The inverse operation of squaring a number. For negative numbers, involves imaginary numbers.
  • Imaginary unit (i): The unit used to express the square root of negative numbers, defined as √(-1).
  • Complex number: A number comprising a real part and an imaginary part, such as a + bi.
  • Prime factorization: Breaking down a number into its prime factors, useful for simplifying square roots.
  • Absolute value: The non-negative value of a number without regard to its sign, used to express magnitudes.

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

▶

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.