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Chapter 10: Problem 23

A student who was directed to find \(\sqrt{-4}\) incorrectly obtained -2 . WHAT WENT WRONG? Give the correct answer.

Short Answer

Expert verified
The correct answer is \( 2i \), not -2.

Step by step solution

01

Understand what \( \sqrt{-4} \) represents

The square root of a negative number involves complex numbers. Note that there is no real number whose square is negative, thus \( \sqrt{-4} \) requires us to use the imaginary unit \( i \), where \( i\) is defined as \sqrt{-1} \.
02

Express -4 in terms of \(i\)

Rewrite -4 as \( -4 = 4 \cdot (-1)\).
03

Apply the square root to both parts

Use the properties of square roots to split this into \( \sqrt{4 \cdot (-1)}\ = \sqrt{4} \cdot \sqrt{-1} \).
04

Simplify the square roots

Since \( \sqrt{4} = 2 \), and \sqrt{-1} = i \, we get \( 2 \cdot i = 2i \).
05

Conclude the answer

Thus, the correct answer to \sqrt{-4} \ is \( 2i \), not -2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square roots of negative numbers
Taking the square root of a negative number can be tricky because no real number squared gives a negative result. To handle square roots of negative numbers, we use complex numbers. Complex numbers combine real and imaginary parts, allowing for more flexibility in mathematics.
Imaginary unit i
The imaginary unit, denoted by \( i \), is defined as \( i = \sqrt{-1} \). This definition helps us work with square roots of negative numbers. When you see \( i \), know it's representing the square root of -1.
Here are a few crucial points to remember:
  • \( i^2 = -1 \)
  • \( i^3 = i \cdot i^2 = -i \)
  • \( i^4 = 1 \)
    These properties often come handy when simplifying expressions involving \( i \).
Simplifying complex numbers
Let鈥檚 learn how to simplify complex numbers using the example from the exercise.
First, consider the original problem \( \sqrt{-4} \). As explained, we cannot find a real number whose square is -4. Instead, we break it down:
  • Rewrite -4 as 4 multiplied by -1: \( -4 = 4 \cdot (-1) \).
  • Use properties of square roots to separate this: \( \sqrt{4 \cdot (-1)} = \sqrt{4} \cdot \sqrt{-1 } \).
  • Next, simplify each part: \( \sqrt{4} = 2 \) and \( \sqrt{-1} = i \).
Putting it all together, you get \( 2 \cdot i = 2i \). So, the correct answer is \( 2i\), not \(-2\). Remember, combining both real and imaginary parts correctly is essential to solve such problems accurately.

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