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 vehicle design and finance. Here, we will discuss the square root of -164.
The square root is the inverse of the square of a number. Since -164 is a negative number, its square root involves imaginary numbers. The square root of -164 is expressed in both radical and exponential form. In the radical form, it is expressed as √(-164), whereas (-164)^(1/2) in the exponential form. The square root of a negative number is an imaginary number, which can be expressed as i√164, where i is the imaginary unit with the property that i² = -1.
To find the square root of a negative number like -164, we need to recognize that it involves the imaginary unit i. The prime factorization and long division methods do not apply directly to negative numbers since they yield real numbers. However, we can find the square root of the positive counterpart and then multiply it by i. Let's explore the steps:
1. Find the square root of 164 using the usual methods.
2. Multiply the result by i to account for the negative sign.
The product of prime factors is the prime factorization of a number. Here’s how 164 is broken down into its prime factors:
Step 1: Finding the prime factors of 164 Breaking it down, we get 2 x 2 x 41: 2² x 41
Step 2: Pair the prime factors. Since 164 is not a perfect square, the digits of the number can’t be grouped into pairs for complete pairs.
Thus, the square root of 164 is expressed as 2√41, and for -164 as i(2√41).
The long division method is typically used for finding the square roots of positive non-perfect square numbers. Here’s how to find the square root of 164 using the long division method, then apply the result to -164:
Step 1: Group the numbers from right to left. For 164, group as 64 and 1.
Step 2: Find n whose square is 1. Here, n is 1 because 1 x 1 = 1. Subtract 1 from 1, resulting in a remainder of 0.
Step 3: Bring down 64, the new dividend. Add the previous divisor 1 to itself to get 2, the new divisor.
Step 4: Find n such that 2n x n ≤ 64. Here, n is 8 because 28 x 8 = 224, and 224 is less than 640.
Step 5: Subtract 224 from 640 to get a remainder of 416 and continue the process to get more decimal places. Once you find the square root of 164, multiply the result by i for -164.
The approximation method is another way to find square roots. Here’s how to find the square root of 164 using the approximation method:
Step 1: Identify the closest perfect squares to 164. The nearest perfect squares are 144 (12²) and 169 (13²). Thus, √164 is between 12 and 13.
Step 2: Use the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying it: (164 - 144) / (169 - 144) = 20 / 25 = 0.8 Add the decimal to the lower bound: 12 + 0.8 = 12.8 For -164, multiply the result by i to get i(12.8).
Students often make mistakes while finding square roots, such as forgetting about the imaginary unit when dealing with negative numbers. Let’s explore some common mistakes in detail.
If Max wants to find the result of i times the square root of 138, what will it be?
The result is approximately 11.75i.
To find the result, calculate the square root of 138, which is approximately 11.75, and multiply by i.
A square has an area of -164 square units. What is the side length in terms of imaginary numbers?
The side length is i√164.
The side length of a square is the square root of its area.
For negative areas, use i to denote the imaginary part: i√164.
Calculate i√164 x 3.
The result is approximately 38.4i.
First, find the square root of 164, which is approximately 12.8, then multiply by 3 and add the imaginary unit: 12.8 x 3 = 38.4, thus 38.4i.
What is the square root of (138 - 2) in terms of imaginary numbers?
The square root is approximately 11.66i.
First, find the square root of (138 - 2) = 136, which is approximately 11.66.
Thus, the square root of the negative is 11.66i.
If a rectangle has a length of i√138 units and a width of 38 units, what is its perimeter in terms of imaginary numbers?
The perimeter is approximately 77.48 + 11.66i units.
Perimeter of the rectangle = 2 × (length + width), where length = i√138 ≈ 11.66i, width = 38.
Thus, the perimeter = 2 × (11.66i + 38) = 77.48 + 23.32i.
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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