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Last updated on May 26th, 2025

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Square Root of -72

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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.

Square Root of -72 for Australian Students
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What is 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.

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Finding the Square Root of -72

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:

 

  • Identify the positive part of the number, which is 72.
  • Find the square root of 72.
  • Multiply the result by 'i'.
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Square Root of -72 by Prime Factorization Method

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.

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Square Root of -72 by Imaginary Number Concept

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.

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Understanding Complex Numbers

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.

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Common Mistakes and How to Avoid Them in the Square Root of -72

When dealing with square roots of negative numbers, understanding the role of imaginary numbers is crucial. Here are common mistakes students make:

Mistake 1

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Ignoring the Imaginary Unit

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Students often forget to include the imaginary unit 'i' when calculating the square root of a negative number.

For example, forgetting 'i' in √-72 leads to an incorrect result. Always remember: √-72 = 6√2 * i.

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Square Root of -72 Examples

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Problem 1

How would you express the area of a square with side length √-72?

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The area would be expressed as an imaginary number, specifically -72 square units.

Explanation

Area = (side length)^2 = (√-72)^2 = -72i^2 = 72, considering i^2 = -1.

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Problem 2

If a rectangle has one side √-72 and another side of 10, what is the perimeter?

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The perimeter is not a real number, as it involves imaginary numbers.

Explanation

Perimeter = 2(length + width) = 2(√-72 + 10) = 2(6√2 * i + 10), which is a complex expression.

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Problem 3

What is the value of (√-72)²?

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The value is 72.

Explanation

(√-72)² = -72i² = 72, because i² = -1.

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Problem 4

Find (√-72) * (√-8).

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The result is 24i.

Explanation

(√-72) * (√-8) = 6√2 * i * 2√2 * i = 12 * 2 * i² = 24(-1) = -24.

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Problem 5

Calculate the modulus of the complex number 6√2 * i.

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The modulus is 6√2.

Explanation

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.

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FAQ on Square Root of -72

1.What is √-72 in its simplest form?

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2.What are the prime factors of 72?

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3.What is a complex number?

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4.Why is the square root of a negative number imaginary?

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5.Is 72 a perfect square?

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6.How does learning Algebra help students in Australia make better decisions in daily life?

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7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of -72?

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8.How do technology and digital tools in Australia support learning Algebra and Square Root of -72?

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9.Does learning Algebra support future career opportunities for students in Australia?

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Important Glossaries for the Square Root of -72

  • Imaginary Unit: The imaginary unit 'i' is defined as √-1, and is used to express the square roots of negative numbers.
     
  • Complex Number: A complex number is a number that includes both a real part and an imaginary part, written in the form a + bi.
     
  • Modulus: The modulus of a complex number a + bi is the distance from the origin to the point (a, b) in the complex plane, calculated as √(a² + b²).
     
  • Prime Factorization: Prime factorization involves expressing a number as the product of its prime factors, useful for simplifying square roots of positive numbers.
     
  • Perfect Square: A perfect square is a number that is the square of an integer. 72 is not a perfect square, hence its square root is not an integer.
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About BrightChamps in Australia

At BrightChamps, we believe algebra is more than symbols—it opens doors to endless opportunities! Our mission is to help children all over Australia gain important math skills, focusing today on the Square Root of -72 with a special emphasis on understanding square roots—in a lively, fun, and easy-to-grasp way. Whether your child is calculating the speed of a roller coaster at Luna Park Sydney, tracking cricket match scores, or managing their allowance for the newest gadgets, mastering algebra gives them the confidence to tackle everyday problems. Our interactive lessons make learning both simple and enjoyable. Since children in Australia learn in various ways, we adapt our approach to fit each learner’s style. From Sydney’s vibrant streets to the stunning Gold Coast beaches, BrightChamps brings math to life, making it relevant and exciting throughout Australia. Let’s make square roots a joyful part of every child’s math journey!
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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.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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