111 LearnersLast updated on May 26th, 2025
Square Root of -175
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.
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.
Mistake 1
Ignoring the Imaginary Unit
When dealing with negative numbers under a square root, students often forget to include the imaginary unit, i. Remember, √(-1) = i.
For example, √(-9) should be written as 3i, not just 3.
Mistake 2
Incorrect Simplification
Another mistake is incorrect simplification of expressions involving square roots. Ensure correct factorization and simplification.
For example, √(-20) should be simplified to 2i√5, not 4i.
Mistake 3
Confusing Real and Imaginary Parts
Students sometimes confuse real and imaginary components when simplifying. Remember, the real part of √(-175) is zero; the imaginary part is 5√7.
Mistake 4
Misapplying Square Root Properties
Be careful not to apply properties of real numbers to imaginary numbers.
For example, √(a×b) = √a×√b only applies if a and b are both non-negative.
Mistake 5
Overlooking Complex Conjugates
When working with complex numbers, it’s important to understand complex conjugates. For instance, the square root of a negative number will always have a complex conjugate.
Square Root of -175 Examples
Problem 1
Can you help Max find the magnitude of a vector if its component along one axis is √(-175)?
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.
Problem 2
Calculate √(-175) × 2.
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.
Problem 3
What is the square of √(-175)?
The square is -175.
Explanation
Squaring the square root returns the original number: (√(-175))^2 = -175.
FAQ on Square Root of -175
1.What is √(-175) in terms of i?
2.What are the prime factors of 175?
3.Is the square root of -175 a real number?
4.What is the absolute value of √(-175)?
5.How do you express √(-175) using imaginary numbers?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of -175?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of -175?
9.Does learning Algebra support future career opportunities for students in Australia?
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.
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Jaskaran Singh Saluja
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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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