Problem 36 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$\sqrt{-63}$$

Step-by-Step Solution

Answer

\(3i \sqrt{7}\)

Step 1: Identify the Imaginary Component

Since we are dealing with the square root of a negative number, we recognize that the imaginary unit, \(i\), where \(i^2 = -1\), will be part of the solution. Any negative inside a square root can be expressed as the square root of the positive number multiplied by \(i\). Thus, \(\sqrt{-63} = i \sqrt{63}\).

Step 2: Simplify the Radical

Now, simplify \(\sqrt{63}\) into its prime factors to make it simpler. Notice that \(63 = 9 \times 7\). Since \(9\) is a perfect square, \(\sqrt{9} = 3\). Then: \[\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3 \sqrt{7}\]

Step 3: Combine the Components

Now, multiply the \(i\) with the simplified expression from the radical step: \(i \times 3 \sqrt{7}\). This gives the simplified form: \(3i \sqrt{7}\).

Key Concepts

Complex Numbers

Complex numbers are a fundamental part of algebra that extends the concept of real numbers. When you come across numbers that include the square root of a negative number, you're venturing into the world of complex numbers.
A complex number is typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The term \(bi\) represents the imaginary part of the complex number, and in this part, \(i\) is the imaginary unit. Here, \(i\) is defined by the property that \(i^2 = -1\).

• The real part "\(a\)" is just like any number you're used to.
• The imaginary part "\(bi\)" accounts for those instances where we have to deal with square roots of negative numbers.

When combined, both parts can help solve equations that otherwise have no real solutions. For instance, equations like \(x^2 + 1 = 0\) are solvable when using complex numbers.

Simplifying Radicals

When you simplify radicals, the goal is to express the square root in the simplest form. A radical expression can be simplified by determining whether any factors of the radicand (the number under the square root symbol) are perfect squares.
Here's a step-by-step breakdown of simplifying radicals:

• Start by factoring the number inside the radical symbol into its prime factors.
• Identify any perfect squares among these factors. A perfect square is a number like 4, 9, 16, etc., because they are squares of integers (2, 3, 4, etc.).
• Extract the square root of any perfect square factors outside the radical.

For example, to simplify \(\sqrt{63}\), note that \(63 = 9 \times 7\). Here, 9 is a perfect square, so \(\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}\).
This process allows for an equivalent expression that is typically more manageable or interpretable.

Square Roots of Negative Numbers

When you encounter the square root of a negative number, it might seem unsolvable at first. However, this is where the imaginary unit \(i\) becomes very useful.
The square root of a negative number can be expressed using \(i\). For example, \(\sqrt{-63}\) can initially be expressed as \(i\sqrt{63}\).
Here's how to think about it:

• Write \(\sqrt{-a}\) as \(i\sqrt{a}\), where \(a\) is a positive number. This is done because \(i^2 = -1\).

By converting negative square roots into products involving \(i\), we effectively create a way of expressing these otherwise non-real numbers. Take \(\sqrt{-4}\), for example. This can be rewritten as \(i\sqrt{4}\), which equals to \(2i\). This process allows for managing and simplifying expressions in algebra more effectively.