Problem 14 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ -64^{1 / 2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(8i\).

Step 1: Extract the Base

Identify the base of the expression, which in this case is -64. The task is to rewrite the expression using radical notation based on this base.

Step 2: Apply Radical Notation

Recognize that the expression (-64)^{1 / 2} is equivalent to the square root of -64 using radical notation. Therefore, the expression becomes \(\sqrt{-64}\).

Step 3: Simplify Using Imaginary Numbers

Realize that the square root of a negative number involves imaginary numbers. Rewrite \(\sqrt{-64}\) as \(\sqrt{64} \times \sqrt{-1}\), which simplifies to \(8\times i\) because \(\sqrt{64} = 8\) and \(\sqrt{-1} = i\).

Key Concepts

Imaginary Numbers

Imaginary numbers come into play when we try to find the square roots of negative numbers. They are represented with the imaginary unit, denoted by 'i'. This imaginary unit is defined as the square root of -1, so \(i^2 = -1\). Imaginary numbers extend the real number line into a new dimension, allowing us to find roots of negative quantities.

Whenever you encounter \(\sqrt{-a}\) (where \(a\) is a positive number), it can be rewritten using \(i\) as \(\sqrt{a} \times i\). This means \(\sqrt{-64}\) equals \(\sqrt{64} \times \sqrt{-1} = 8 \times i\), simplifying expressions and creating a new set of numbers to work with, known as complex numbers. Remember, complex numbers have both real and imaginary parts. They are written as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

Square Root

The square root is a mathematical operation that finds the original number which was squared to get a given value. It's written using the radical symbol \(\sqrt{}\). For example, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\).

However, when dealing with negative numbers under the radical, such as \(-64\), we can't find real number roots since a real number times itself will never result in a negative number. To overcome this, we use imaginary numbers to express the square root. By splitting \(\sqrt{-64}\) into \(\sqrt{64} \times \sqrt{-1}\), we use the rules of square roots to simplify and find the solution \(8 \times i\). This understanding is crucial for handling expressions involving negative numbers under the square root.

Simplifying Expressions

Simplifying expressions involves combining like terms and converting complex expressions into simpler forms. In our case, taking the expression \( (-64)^{1/2} \) and rewriting it with radical notation as \(\sqrt{-64}\) was just the start.

Further simplification involved breaking the expression into \(\sqrt{64} \times \sqrt{-1}\). Recognizing that \(\sqrt{64} = 8\), and the square root of \(-1\) is \(i\), the expression simplifies to \(8 \times i\). This process demonstrated the power of using imaginary numbers to simplify and resolve otherwise complex-looking mathematical problems. Simplifying helps solve problems more efficiently, making complex calculations more manageable.