Problem 8 - Algebra A Combined Function

Question

Write using i notation. $$ \sqrt{-63} $$

Step-by-Step Solution

Answer

\( \sqrt{-63} = 3i\sqrt{7} \).

Step 1: Understand i Notation

In mathematics, the imaginary unit is denoted by \( i \), where \( i^2 = -1 \). This notation allows us to handle square roots of negative numbers.

Step 2: Separate the Negative Sign

Rewrite \( \sqrt{-63} \) as \( \sqrt{-1 \times 63} \). This allows us to separate the negative number part from the square root.

Step 3: Use i Notation

Replace \( \sqrt{-1} \) with \( i \). So, \( \sqrt{-1 \times 63} = i\sqrt{63} \).

Step 4: Simplify \( \sqrt{63} \)

Notice that \( 63 = 9 \times 7 \). So, \( \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7} \).

Step 5: Combine the Expressions

Replace \( \sqrt{63} \) in our expression with its simplified form: \( i \times 3\sqrt{7} \).

Step 6: Final Expression

The final expression in terms of \( i \) notation is \( 3i\sqrt{7} \).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers; they include a real part and an imaginary part. The imaginary unit is designated as \( i \), which is defined by the property \( i^2 = -1 \). This notation gives us the ability to include solutions for equations that involve the square roots of negative numbers, something not possible with just the real numbers alone. Complex numbers are typically represented in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

• Complex numbers are a vital part of mathematics, used to solve quadratic equations with no real roots.
• They also appear in signal processing, control systems, and many other engineering and physics applications.
• When you see \( i \), remember it’s a tool to help deal with the square roots of negative numbers.

Understanding complex numbers opens up a whole world of higher mathematics, providing solutions and techniques that are essential for advanced problem-solving.

Square Roots

The concept of square roots is fundamental in mathematics. A square root of a number \( x \) is a number \( y \) such that \( y^2 = x \). However, real numbers have an important limitation: we cannot take the square root of a negative number without using imaginary numbers.

When dealing with square roots of negative numbers, we use the imaginary unit \( i \) to help. For example:

• To compute \(\sqrt{-63}\), acknowledge that it is equivalent to \(\sqrt{-1 \times 63}\).
• The imaginary unit \( i \) simplifies \(\sqrt{-1}\) to \( i \), giving us \( i\sqrt{63}\).

This transformation allows us to handle the square roots of negative values seamlessly by switching to the realm of complex numbers. It’s a key skill for solving many types of mathematical equations.

Simplification

Simplification is the process of breaking down a complicated expression into simpler parts to make it more manageable. In the context of expressions involving square roots, like \( \sqrt{63} \), we can simplify by factoring.

For \( \sqrt{63} \), notice that \( 63 = 9 \times 7 \). You can rewrite the square root expression as \( \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} \). Since \( \sqrt{9} \) simplifies to 3, the expression becomes \( 3\sqrt{7} \).

• This simplification makes the original square root expression much easier to work with.
• When combined with imaginary notation, as in our example \( 3i\sqrt{7} \), it allows precise handling of otherwise unwieldy mathematical computations.

Through simplification, complex computations become accessible and solvable, a necessary step in both mathematical problem-solving and real-world applications.