Problem 6 - Algebra A Combined Function

Question

Write using i notation. $$ 4 \sqrt{-20} $$

Step-by-Step Solution

Answer

8i\sqrt{5}

Step 1: Rewrite the Radical

The expression given is \(4\sqrt{-20}\). First, identify the negative number inside the square root, \(-20\). Rewriting this, \(\sqrt{-20} = \sqrt{-1 \times 20}\). This utilizes the property that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).

Step 2: Extract the Negative Factor

We need to express \(\sqrt{-1 \times 20}\) in terms of \(i\) notation. The square root of \(-1\) is defined as \(i\). So, \(\sqrt{-1 \times 20} = \sqrt{-1} \times \sqrt{20} = i \times \sqrt{20}\).

Step 3: Simplify the Square Root of 20

Next, simplify \(\sqrt{20}\). Notice that \(20 = 4 \times 5\), making \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2 \sqrt{5}\).

Step 4: Combine All Parts

Now, substitute back into the expression where \(\sqrt{-20} = i \times 2 \sqrt{5}\). This becomes \(2i\sqrt{5}\). Thus, the original expression \(4 \sqrt{-20}\) can be written as \(4 \times 2i \sqrt{5}\).

Step 5: Final Calculation

Multiply out the coefficients: \(4 \times 2 = 8\), so \(4 \sqrt{-20} = 8i \sqrt{5}\). Therefore, the expression in terms of \(i\) notation is \(8i \sqrt{5}\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), serves as a fundamental building block in the world of complex numbers. It is formally defined as the square root of \(-1\). This unique property means that \(i^2 = -1\).

In mathematics, whenever we encounter the square root of a negative number, we utilize \(i\) to handle it. For instance, the square root of \(-4\) is expressed as \(2i\), because \(\sqrt{-4} = \sqrt{-1 \times 4} = \sqrt{-1} \times \sqrt{4} = i \times 2 = 2i\).

The imaginary unit allows us to extend the real number system into the complex number system, where numbers have both real and imaginary parts. This is particularly important for solving equations that do not have solutions within the domain of real numbers.

Square Roots

The square root operation is the opposite of squaring a number. Finding the square root involves determining a number which, when multiplied by itself, gives the original number. Normally, square roots are only defined for non-negative numbers in real numbers because the square of any real number is non-negative.

Here's a quick review of square roots:

• The square root of 16 is 4, because \(4 \times 4 = 16\).
• When working with radicals, like \(\sqrt{9}\), it’s important to simplify wherever possible: \(\sqrt{9} = 3\).

However, in the world of complex numbers, we also deal with square roots of negative numbers by using \(i\). For example, \(\sqrt{-25} = \sqrt{-1 \times 25} = i \times 5 = 5i\). By including \(i\), we are able to express and handle negative square roots.

Properties of Radicals

Radicals involve roots, such as square roots, and come with their own set of properties. These rules are helpful when simplifying expressions that include square roots. Understanding these properties is crucial while working with complex numbers.

Some fundamental properties of radicals include:

• \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\): This property allows us to break down a square root into more manageable pieces. For example, \(\sqrt{36} = \sqrt{9 \times 4} = \sqrt{9} \times \sqrt{4} = 3 \times 2 = 6\).
• \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\): If \(a\) and \(b\) are positive, you can administer the root to both the numerator and the denominator separately.
• Simplifying radicals involves expressing them in their simplest form, such as converting \(\sqrt{20}\) to \(2\sqrt{5}\).

When dealing with negative radicands, like \(-20\) in this case, incorporating the imaginary unit \(i\) becomes necessary: \(\sqrt{-20} = i \sqrt{20} = i \times 2 \sqrt{5}\). Recognizing and using these properties provides clarity and precision in simplifying complex expressions.