Last updated on May 26th, 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.
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.
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.
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.
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.
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.
Can you help Max find the magnitude of a vector if its component along one axis is √(-175)?
The magnitude is 5√7.
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.
Calculate √(-175) × 2.
The result is 10i√7.
First, find the square root of -175, which is 5i√7.
Then multiply by 2: 5i√7 × 2 = 10i√7.
What is the square of √(-175)?
The square is -175.
Squaring the square root returns the original number: (√(-175))^2 = -175.
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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