Problem 38 - A Graphical Approach to College Algebra

Question

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-70}}{\sqrt{-7}}$$

Step-by-Step Solution

Answer

The simplified result is \( \sqrt{10} \).

Step 1: Recognize the Involvement of Complex Numbers

Notice that both the numerator and the denominator involve square roots of negative numbers. Therefore, the result will be a complex number involving imaginary units.

Step 2: Simplify Each Square Root Individually

Write the square roots using imaginary numbers: \( \sqrt{-70} = \sqrt{70} \cdot i \) and \( \sqrt{-7} = \sqrt{7} \cdot i \).

Step 3: Substitute Imaginary Units and Simplify the Fraction

Substitute the expressions from Step 2 back into the fraction to get \( \frac{\sqrt{70} \cdot i}{\sqrt{7} \cdot i} \). Since \( i \) is in both the numerator and the denominator, they cancel each other out.

Step 4: Handle the Fraction with Remaining Real Parts

Now simplify the real part of the fraction: \( \frac{\sqrt{70}}{\sqrt{7}} = \sqrt{\frac{70}{7}} = \sqrt{10} \).

Step 5: State the Final Result

The simplified form of the given expression is \( \sqrt{10} \).

Key Concepts

Imaginary Unit

When dealing with complex numbers, one often encounters the imaginary unit. This special number is denoted by \( i \) and is defined by the property that \( i^2 = -1 \). This might seem odd since the square of a real number is always positive or zero. However, in the realm of complex numbers, \( i \) allows us to work with the square roots of negative numbers, which are undefined in the set of real numbers.

For any negative number \(-a\), the square root can be expressed as \( \sqrt{-a} = \sqrt{a} \cdot i \). This is because \( \sqrt{a} \) is a real number and \( i \) accounts for the negative sign, thanks to its unique property.

• Example: \( \sqrt{-9} = \sqrt{9} \cdot i = 3i \).

Understanding the imaginary unit is key to solving equations and expressions that involve negative square roots.

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra. It involves making a fraction as simple as possible so that the numerator and denominator have no common factors other than 1.

In the context of complex numbers, simplifying fractions might involve canceling out common units, such as the imaginary unit \( i \), across the numerator and denominator. Consider the expression \( \frac{a \cdot i}{b \cdot i} \). Since \( i \) appears in both parts of the fraction, it can be canceled, leaving \( \frac{a}{b} \), which is often easier to work with.

• Cancel common factors: This reduces complexity and simplifies calculations.
• Focus on the real parts if the imaginary components are identical.

Such simplifications are essential in reducing the complexity of problems involving complex numbers.

Square Roots

Square roots are a frequent operation in mathematics, denoting a number which, when multiplied by itself, yields the original number. Square roots appear challenging when they involve negative numbers, which introduce the imaginary unit \( i \).

The essence of simplifying square roots is breaking them down into components that are easier to work with. Consider \( \sqrt{rac{n}{m}} \), which can be simplified to \( \sqrt{n} / \sqrt{m} \) if both \( n \) and \( m \) are positive. This rule applies to real numbers and is useful in simplifying expressions.

• Always check for perfect square factors to simplify the root further.
• Use \( i \) for negative numbers to keep the expression within the complex number system.

Understanding and simplifying square roots paves the way for solving complex expressions easily.