Problem 57 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \frac{\sqrt{-56}}{\sqrt{-7}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(2\sqrt{2}\).

Step 1: Simplify the square roots using imaginary numbers

Determine the square roots of negative numbers using the imaginary unit, where \( i^2 = -1 \). The square root of any negative number \( -x \) can be expressed as \( \sqrt{x} \times i \). Therefore, \( \sqrt{-56} = \sqrt{56} \cdot i \) and \( \sqrt{-7} = \sqrt{7} \cdot i \).

Step 2: Substitute simplified forms into the expression

Replace the original square roots with their expressions in terms of \( i \). The expression becomes: \[\frac{\sqrt{56} \cdot i}{\sqrt{7} \cdot i}\]

Step 3: Cancel the imaginary unit \(i\)

Since \( i \) is in both the numerator and the denominator, it can be canceled out. The expression simplifies to: \[\frac{\sqrt{56}}{\sqrt{7}}\]

Step 4: Simplify the remaining square root fraction

Simplify \( \frac{\sqrt{56}}{\sqrt{7}} \) by dividing the contents under the square roots: \[\sqrt{\frac{56}{7}} = \sqrt{8}\]

Step 5: Simplify \(\sqrt{8}\) further

Simplify \(\sqrt{8}\) by finding factors that are perfect squares: \[\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \cdot \sqrt{2}\]

Key Concepts

Square Roots of Negative Numbers

Understanding square roots is crucial, especially when they involve negative numbers. When dealing with these, we use the imaginary number unit denoted as \( i \), where \( i^2 = -1 \). This is because the square root of a negative number doesn't exist in the set of real numbers. Instead, we express the square root of a negative number \(-x\) as \( \sqrt{x} \times i \). For example, to find \( \sqrt{-56} \), we rearrange it as \( \sqrt{56} \times i \). Likewise, \( \sqrt{-7} \) becomes \( \sqrt{7} \times i \).

The process involves two important steps:

• First, taking the square root of the positive equivalent of the number.
• Then, multiplying by the imaginary unit \( i \).

This method helps us manipulate and simplify equations that include square roots of negative numbers with ease.

Simplification of Algebraic Expressions

After rewriting negative square roots in terms of \( i \), algebraic simplification comes next. The expression \( \frac{\sqrt{56} \cdot i}{\sqrt{7} \cdot i} \) includes \( i \) both in the numerator and the denominator, which allows us to cancel it out, provided \( i eq 0 \). This step streamlines the expression to \( \frac{\sqrt{56}}{\sqrt{7}} \).

Here's what to remember:

• Cancel out like terms in the numerator and denominator.
• Simplify expressions by eliminating the imaginary unit when present in both parts.

By correctly following these steps, you ensure the expression is no longer complicated by extra factors of \( i \), easing further simplification and helping us move to the next steps effectively.

Application of Intermediate Algebra Techniques

Intermediate algebra provides the tools to simplify complicated expressions further. Now that we've reached \( \frac{\sqrt{56}}{\sqrt{7}} \), we can simplify it by dividing under the square root. This yields \( \sqrt{\frac{56}{7}} \), which simplifies to \( \sqrt{8} \).
The final step involves simplifying \( \sqrt{8} \):

• Find factors of 8 that include a perfect square.
• We express 8 as \( 4 \times 2 \), where 4 is a perfect square.
• So, \( \sqrt{8} = \sqrt{4} \cdot \sqrt{2} = 2 \cdot \sqrt{2} \).

Mastering these techniques allows one to effectively manage and simplify expressions, skills particularly useful within algebraic manipulation and essential for higher-level mathematics.