Last updated on May 26th, 2025
The square root is an inverse operation to squaring a number. When dealing with negative numbers, the concept of square roots involves complex numbers. In this context, we will explore the square root of -72.
The square root of a negative number introduces the concept of imaginary numbers. While 72 is not a perfect square, the square root of -72 is expressed using the imaginary unit 'i', where i = √-1. In exponential form, it is expressed as (-72)^(1/2). The square root of -72 is 6√2 * i, which is a complex number because it involves the imaginary unit.
To find the square root of a negative number like -72, we use the concept of imaginary numbers. This involves treating the negative sign separately and finding the square root of the positive part, then multiplying by 'i'. Here are the steps involved:
While prime factorization is typically used for positive integers, we can find the square root of the positive component of -72 first.
Step 1: Find the prime factors of 72.
Breaking it down, we have 2 x 2 x 2 x 3 x 3: 2^3 x 3^2.
Step 2: Pair the prime factors. For perfect squares, every prime factor should appear in pairs. Here, 2^2 and 3^2 can be paired, leaving 2 unpaired.
Step 3: Calculate the square root of 72 as √(2^3 x 3^2) = 6√2.
Step 4: The square root of -72 is 6√2 * i.
The imaginary number concept is used for negative square roots. Here's how to find the square root of -72:
Step 1: Separate the negative sign and express it with 'i'.
Step 2: Calculate the square root of 72, which is 6√2.
Step 3: Combine with 'i': The square root of -72 is 6√2 * i.
Complex numbers consist of a real part and an imaginary part. For the square root of -72, the result is purely imaginary since the real part is zero. Imaginary numbers are useful in various fields, including engineering and physics.
When dealing with square roots of negative numbers, understanding the role of imaginary numbers is crucial. Here are common mistakes students make:
How would you express the area of a square with side length √-72?
The area would be expressed as an imaginary number, specifically -72 square units.
Area = (side length)^2 = (√-72)^2 = -72i^2 = 72, considering i^2 = -1.
If a rectangle has one side √-72 and another side of 10, what is the perimeter?
The perimeter is not a real number, as it involves imaginary numbers.
Perimeter = 2(length + width) = 2(√-72 + 10) = 2(6√2 * i + 10), which is a complex expression.
What is the value of (√-72)²?
The value is 72.
(√-72)² = -72i² = 72, because i² = -1.
Find (√-72) * (√-8).
The result is 24i.
(√-72) * (√-8) = 6√2 * i * 2√2 * i = 12 * 2 * i² = 24(-1) = -24.
Calculate the modulus of the complex number 6√2 * i.
The modulus is 6√2.
The modulus of a complex number a + bi is √(a² + b²).
Here, a = 0 and b = 6√2, so modulus = √(0² + (6√2)²) = 6√2.
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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