Problem 25 - Intermediate Algebra

Question

$$ \frac{\sqrt{-80}}{\sqrt{-10}} $$

Step-by-Step Solution

Answer

The simplified result is \( 2 \sqrt{2} \).

Step 1: Simplify Negative Radicals Using Imaginary Numbers

The given expression involves square roots of negative numbers, which we can handle using imaginary numbers. Recall that \( \sqrt{-1} = i \), where \( i \) is the imaginary unit. Rewrite each square root as follows: \[ \sqrt{-80} = \sqrt{-1} \cdot \sqrt{80} = i \sqrt{80} \] \[ \sqrt{-10} = \sqrt{-1} \cdot \sqrt{10} = i \sqrt{10} \].

Step 2: Substitute Simplified Radicals into the Fraction

Now that we have the square roots in terms of \( i \), substitute these back into the original fraction: \[ \frac{i \sqrt{80}}{i \sqrt{10}} \].

Step 3: Simplify the Fraction by Canceling \( i \)

Since \( i \) is present in both the numerator and the denominator, we can simplify the fraction by canceling \( i \):\[ \frac{i \sqrt{80}}{i \sqrt{10}} = \frac{\sqrt{80}}{\sqrt{10}} \].

Step 4: Simplify the Radical Fraction

Use the property \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) to simplify the fraction:\[ \frac{\sqrt{80}}{\sqrt{10}} = \sqrt{\frac{80}{10}} = \sqrt{8} \].

Step 5: Simplify the Final Radical

Now simplify \( \sqrt{8} \). Recognize that \( 8 = 4 \cdot 2 \) and \( \sqrt{4} = 2 \):\[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2} \].

Key Concepts

Imaginary Numbers

Imaginary numbers are a fascinating concept in mathematics, providing a solution to equations that don't have solutions in the real number system. The term comes from the introduction of the "imaginary unit," denoted as \(i\), which is defined as \( i = \sqrt{-1} \). This might seem like an unusual definition, but it allows us to solve problems involving negative square roots, which was impossible before.

Imaginary numbers are crucial in various fields like engineering and physics, where they help in analyzing phenomena such as electrical circuits and wave mechanics.

• The imaginary unit \(i\) follows specific rules: \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\) (and then it repeats).
• When working with square roots of negative numbers, you express them using \(i\). For example, \(\sqrt{-4} = 2i\).

Understanding this concept is key to simplifying expressions involving square roots of negative numbers, as it transforms them into manageable forms.

Properties of Exponents

Properties of exponents allow us to perform operations on powers and roots efficiently. These rules are the foundation for manipulating expressions and solving equations with exponents. By remembering and applying these properties, you can simplify many mathematical expressions with ease.

Consider these essential properties:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^n = a^n \times b^n\)
• Power of a Fraction: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Understanding and applying these rules will simplify working with both whole numbers and radicals. Use them to breakdown expressions to their simplest forms, like when simplifying \(\frac{\sqrt{80}}{\sqrt{10}}\) to \(\sqrt{8}\).

Square Roots

Square roots are numbers that, when multiplied by themselves, result in the original number. The square root of a number \(x\) is denoted as \(\sqrt{x}\). It's important to remember that every positive real number has two square roots: one positive and one negative, though by convention, \(\sqrt{x}\) refers to the positive root.

Square roots are often encountered in geometric and algebraic problems. Here are some ways to simplify them:

• Breaking Down Radicals: Decompose the number into prime factors. Example: \(\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2}\).
• Rationalizing Denominators: Multiply the numerator and denominator by a suitable square root to make the denominator a rational number.
• Combining and Simplifying: Use \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) and \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) to simplify expressions.

By mastering these methods, you can efficiently handle expressions similar to simplifying \(\sqrt{8}\) from the original exercise.

Complex Numbers

Complex numbers bring a whole new dimension to mathematics by combining real and imaginary numbers. A complex number has the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Complex numbers allow for the solution of all polynomial equations and have applications across physics and engineering.

Here are a few key properties of complex numbers:

• The set of complex numbers includes all real and imaginary numbers.
• Addition and subtraction are performed component-wise: \((a+bi) + (c+di) = (a+c) + (b+d)i\).
• Multiplication involves distributing the terms: \((a+bi) \times (c+di) = (ac-bd) + (ad+bc)i\).
• The complex conjugate \(\overline{z}\) of \(z = a+bi\) is \(a-bi\). It is used to divide complex numbers or find their modulus \(|z| = \sqrt{a^2 + b^2}\).

Understanding complex numbers helps in manipulating and simplifying mathematical expressions involving square roots of negative numbers, as they account for both real and imaginary components.