Problem 26 - Intermediate Algebra

Question

Express each number in terms of \(i\). $$ 6 \sqrt{-49} $$

Step-by-Step Solution

Answer

The expression is \(42i\).

Step 1: Understanding the Problem

We are asked to express the given number \(6 \sqrt{-49}\) in terms of \(i\), which denotes the imaginary unit. Recall that \(i\) is defined as the square root of \(-1\).

Step 2: Rewriting Negative Square Root

The expression \(\sqrt{-49}\) can be rewritten by separating the negative sign. We have \(\sqrt{-49} = \sqrt{-1 \times 49} = \sqrt{-1} \times \sqrt{49}\).

Step 3: Expressing Square Root of -1 as i

Since \(\sqrt{-1} = i\), replace \(\sqrt{-1}\) with \(i\). The expression becomes \(\sqrt{-49} = i \times \sqrt{49}\).

Step 4: Calculating the Square Root of 49

The square root of 49 is \(7\) because \(7^2 = 49\). So we can substitute \(\sqrt{49}\) with \(7\) in our expression. Now we have \(\sqrt{-49} = 7i\).

Step 5: Final Expression of the Given Number

Combine the results from the previous steps and multiply by 6 as originally given. Thus, \(6 \sqrt{-49} = 6 \times 7i = 42i\).

Key Concepts

Understanding the Imaginary Unit

In mathematics, when working with complex numbers, you often encounter the imaginary unit, denoted by \(i\). The imaginary unit is defined by the equation \(i = \sqrt{-1}\). It essentially serves as the foundation for working with square roots of negative numbers, which don't exist in the set of real numbers. This concept helps us expand into the realm of complex numbers, a critical topic in algebra and higher-level mathematics.

• \(i^2 = -1\): Understanding this property is crucial. It implies that squaring the imaginary unit flips the sign.
• \(i^3 = -i\): This follows from \(i^2 \cdot i = -1 \cdot i = -i\).
• \(i^4 = 1\): This happens because \(i^2 = -1\) and \((-1)^2 = 1\).

These cyclical properties help when simplifying expressions involving complex numbers. Remember, transitioning to using \(i\) lets us manage the impossible square roots of negative numbers much more easily.

Square Roots and Their Rules

Square roots are fundamental in mathematics as they allow us to "reverse" the squaring process. However, when you have square roots involving negative numbers, you enter the realm of complex numbers. For example, the square root of a negative number, such as \(-49\), can be broken down for simplification.
To tackle this, you separate the negative factor, \(-1\), from the number. So, \[ \sqrt{-49} = \sqrt{-1 \times 49} = \sqrt{-1} \times \sqrt{49} \]

• \(\sqrt{-1}\) becomes \(i\), our imaginary unit.
• \(\sqrt{49}\) returns the real number 7.

Therefore, when simplified, \(\sqrt{-49} = 7i\). This method is incredibly helpful for understanding how to break down complex expressions using square roots.

Simplifying Expressions Involving Imaginary Numbers

Simplifying expressions with imaginary numbers might look daunting at first, but the steps become natural with practice. Let's simplify the expression \(6\sqrt{-49}\). First, we rewrite the square root as \(7i\) because we previously determined \(\sqrt{-49} = 7i\). Then, multiply by 6 (as in the original problem). Compute:\[ 6\sqrt{-49} = 6 \times 7i \]By calculating this, we get \(42i\). It's essential to maintain clarity at each step:

• First, resolve the square root dealing with negatives by applying the imaginary unit \(i\).
• Next, calculate any remaining multiplications.
• Finally, assemble the results, ensuring to follow any algebraic rules of imaginary numbers you use along the way.

By following this method, handling expressions with \(i\) becomes more straightforward and systematic, eliminating potential confusion.