Problem 31 - Intermediate Algebra

Question

Write each of the following in terms of \(i\) and simplify. $$ \sqrt{-\frac{16}{25}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4}{5}i\).

Step 1: Identify the Negative Sign

The expression involves the square root of a negative number. Recall that \( \sqrt{-1} = i \), where \(i\) is the imaginary unit.

Step 2: Break Down the Fraction

Separate the negative sign and the fractional part. Rewrite \ \(\sqrt{-\frac{16}{25}}\) as \(-1 \times \frac{16}{25}\). This gives us \(\sqrt{-1} \times \sqrt{\frac{16}{25}}\).\

Step 3: Simplify the Fractional Square Root

Simplify \(\sqrt{\frac{16}{25}}\) as the square root of the numerator over the square root of the denominator: \[\frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}\]

Step 4: Combine with Imaginary Unit

Combine \(\sqrt{-1}\) which is \(i\) with the simplified fraction: \[i\times \frac{4}{5} = \frac{4}{5}i\]

Step 5: Final Simplification

The final simplified form of \(\sqrt{-\frac{16}{25}}\) in terms of \(i\) is \(\frac{4}{5}i\).

Key Concepts

Complex Numbers

Complex numbers are an exciting extension of traditional numbers! They incorporate both real and imaginary components. You'll often see them written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

The imaginary unit \(i\) is the heart of complex numbers. It represents the square root of \(-1\). This means \(i^2 = -1\). When dealing with real numbers, it's impossible to take the square root of a negative number, but using \(i\), we can broaden our mathematical toolkit.

**Why Complex Numbers?**

Complex numbers are used in advanced topics like engineering and physics. They help with solving equations that have no real solutions. They are crucial for signal processing and solving oscillation problems.

• An equation like \(x^2 + 1 = 0\) can be solved in the complex plane.
• It offers a full range of number systems, including real, imaginary, and combinations.

Simplifying Radicals

Simplifying radicals involves rewriting them in their simplest form. This process becomes straightforward with practice and helps in solving equations more efficiently.

The goal is to break down the expression under the square root, using the product or quotient property of square roots.

**How Do You Simplify Radicals?**

When you see a fraction inside a radical, separate the numerator and the denominator as independent radicals. For instance:

• For \(\sqrt{\frac{16}{25}}\), break it into \(\frac{\sqrt{16}}{\sqrt{25}}\).
• Calculate \(\sqrt{16} = 4\) and \(\sqrt{25} = 5\).

The radical expression is now simplified to \(\frac{4}{5}\).

This skill is essential in algebra, helping to simplify more complex expressions and solve radical equations.

Fractional Square Roots

Fractional square roots might seem tricky at first, but they are just steps in expressing and simplifying roots of fractions.

When you calculate the square root of a fraction, it's done by determining the square root of the numerator and the square root of the denominator separately.

**Step-by-Step Approach**

Consider \(\sqrt{\frac{16}{25}}\). Here's a step-by-step breakdown:

• Find \(\sqrt{16}\). Since 16 is a perfect square, \(\sqrt{16} = 4\).
• Find \(\sqrt{25}\). 25 is also a perfect square, so \(\sqrt{25} = 5\).
• Now, combine these results to form the fractional square root: \(\frac{4}{5}\).

Fractional square roots allow for easier computation and simplification of complex problems.

They provide clarity in representing values derived from fractional expressions. This is commonly used in mathematics to simplify and approach problems more directly, and also helps with understanding roots better across different scenarios.