Problem 49 - Elementary Algebra

Question

\(\sqrt{-49}\)

Step-by-Step Solution

Answer

7i

Step 1: Recognize the Negative Sign Inside the Square Root

Identify that the square root includes a negative number. The expression is \(\root{-49}\), meaning you need to deal with the negative sign first.

Step 2: Use the Imaginary Unit

Remember that \( \root{-1} \) is defined as the imaginary unit \(i\). Therefore, \( \root{-49} \) can be expressed as \( \root{-1} \times \root{49} \). This simplifies to \(i \root{49} \).

Step 3: Calculate the Square Root of 49

The square root of 49 is 7 because \(\root{49} = 7 \).

Step 4: Combine the Results

Combine the results from the previous steps. Therefore, \( \root{-49} \) simplifies to \( 7i \).

Key Concepts

imaginary unit

When dealing with the square root of a negative number, you encounter the concept of the 'imaginary unit.' The imaginary unit is represented by the symbol 'i.' It is defined by the equation \(\root{-1} = i\).

This definition allows us to extend the number system to include numbers that are not found on the traditional number line. Instead, these numbers exist in what's called the complex plane.

So, whenever you come across a square root of a negative number, you can use 'i' to represent the square root of -1.

complex numbers

A complex number is a combination of a real number and an imaginary number. It is usually written in the form \(a + bi\), where:

• \text{a is the real part}
• \text{bi is the imaginary part}

For example, in the sum \(3 + 4i\), 3 is the real part and 4i is the imaginary part.

Complex numbers are essential for various fields in science and engineering because they allow for solutions to equations that don't have real-number solutions. Thus, they provide a more complete understanding of various mathematical and physical phenomena.

square roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 49 is 7 because \(7 \times 7 = 49\).

However, negative numbers don't have real square roots because a negative number times itself is always positive. To resolve this, we use the imaginary unit 'i'. For example, the square root of -49 is written as \(\root{-49}\). By using the imaginary unit, we can express it as \(7i\).
This approach allows us to solve equations that wouldn't otherwise have solutions in the realm of real numbers.

mathematical operations

Mathematical operations with complex numbers are very similar to those with real numbers but with the added complexity of the imaginary unit 'i'. Here are a few crucial points:

• Addition: \((a + bi) + (c + di) = (a + c) + (b + d)i\)
• Subtraction: \((a + bi) - (c + di) = (a - c) + (b - d)i\)
• Multiplication: Multiply terms as you would with polynomials, then replace \(i^2\) with -1 like so: \((a + bi)(c + di) = ac + adi + bci + bdi^2\) which simplifies to \(ac + adi + bci - bd = (ac - bd) + (ad + bc)i\)
• Division: When dividing, you'll often use the conjugate of the denominator to rationalize the expression. For example, for \(\frac{a+bi}{c+di}\), you multiply numerator and denominator by the conjugate \(c-di\).

Understanding these operations allows you to manipulate complex numbers in equations easily and solve problems in various fields, from electrical engineering to quantum physics.