Problem 29 - College Algebra Essentials

Question

In Exercises \(29-44\), perform the indicated operations and write the result in standard form. $$ \sqrt{-64}-\sqrt{-25} $$

Step-by-Step Solution

Answer

The result in standard form is \(3i\).

Step 1: Calculate the square root of \(-64\)

Since we're dealing with a negative number under the square root we have to employ \(i\). The square root can be calculated as follows: \(\sqrt{-64}\) = \(\sqrt{64} \cdot \sqrt{-1}\) = \(8i\).

Step 2: Calculate the square root of \(-25\)

As in the previous step, we have to use \(i\). The calculation is as follows: \(\sqrt{-25}\) = \(\sqrt{25} \cdot \sqrt{-1}\) = \(5i\),

Step 3: Perform the subtraction

Subtract the result from step 2 from the result in step 1: \(8i - 5i = 3i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It's defined by the property that \(i^2 = -1\). This means that \(i\) is the square root of \(-1\), and this allows us to work with the square roots of negative numbers, which are not possible with real numbers alone. Understanding \(i\) allows us to perform operations on complex numbers, which include numbers beyond the regular real number system.

• \(i\) is used to indicate the presence of an imaginary number.
• It serves as a placeholder for the square root of negative numbers.
• \(i\) follows certain arithmetic rules: if \(i^2 = -1\), then \(i^3 = -i\) and \(i^4 = 1\), continuing to cycle through these values.

These properties make it possible to solve equations and perform operations involving the square roots of negative numbers by expressing them with the imaginary unit \(i\). This is especially useful in fields like engineering, physics, and computer science, where complex numbers provide meaningful interpretations of real-world phenomena.

Square Root of Negative Numbers

Taking the square root of negative numbers can be tricky because traditional real numbers do not allow for it. This is where the imaginary unit \(i\) comes in handy. The square root of any negative number can be rewritten using \(i\). For example, to find the square root of \(-64\), you use the formula \[\sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8 \cdot i = 8i\]To break it down:

• Calculate the absolute value's square root: \(\sqrt{64} = 8\).
• Incorporate the imaginary unit for the negative part: \(\sqrt{-1} = i\).
• Combine these parts to express the result as an imaginary number, \(8i\).

Similarly, for \(-25\), the square root is \[\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5 \cdot i = 5i\]This method transforms a seemingly impossible calculation into a straightforward one using the imaginary unit. Once rewritten in this form, subtraction, addition, and other arithmetic operations are easily performed.

Standard Form of Complex Numbers

Complex numbers are normally expressed in the standard form, which is \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This makes them a combination of real and imaginary parts, providing a flexible way of representing all numbers in the complex plane.

• The real part \(a\) is any regular real number, which sits on the horizontal axis of the complex plane.
• The imaginary part \(bi\) involves the imaginary unit and resides on the vertical axis of the complex plane.
• Every complex number has a unique representation in this form.

In our problem, by calculating and reducing the expression \(\sqrt{-64} - \sqrt{-25}\), you end up with \[8i - 5i = 3i\]Here, the real part is \(0\) and the imaginary part is \(3i\), so the result \(3i\) is already in standard form. This shows how complex arithmetic simplifies seemingly daunting expressions and situates them neatly within the context of the complex number system.