Problem 30 - Algebra 2 and Trigonometry

Question

In \(19-34,\) write each sum or difference in terms of \(i\) $$ -2+\sqrt{-16}+7-\sqrt{-49} $$

Step-by-Step Solution

Answer

The sum simplifies to \(5 - 3i\).

Step 1: Recognize Imaginary Numbers

In this problem, you'll need to deal with square roots of negative numbers. The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). This means \(\sqrt{-16}\) can be rewritten using \(i\).

Step 2: Simplify \(\sqrt{-16}\)

Since \(\sqrt{-16} = \sqrt{16} \times \sqrt{-1}\), it can be rewritten as \(4i\), because \(\sqrt{16} = 4\) and \(\sqrt{-1} = i\).

Step 3: Simplify \(\sqrt{-49}\)

Similarly, \(\sqrt{-49} = \sqrt{49} \times \sqrt{-1}\) can be rewritten as \(7i\), because \(\sqrt{49} = 7\).

Step 4: Substitute Back into the Expression

Replace \(\sqrt{-16}\) with \(4i\) and \(\sqrt{-49}\) with \(7i\) into the original expression: \(-2 + 4i + 7 - 7i\)

Step 5: Combine Like Terms

Combine the real terms and the imaginary terms separately:- Real terms: \(-2 + 7 = 5\)- Imaginary terms: \(4i - 7i = -3i\)So the expression simplifies to \(5 - 3i\).

Key Concepts

Understanding Complex Numbers

Complex numbers are an essential topic in mathematics, extending the concept of numbers beyond real numbers. They consist of a real part and an imaginary part. The standard form for a complex number is given as \(a + bi\), where:

• \(a\) is the real part, and
• \(b\) is the coefficient of the imaginary unit \(i\).

The imaginary unit \(i\) is defined as the square root of \(-1\), which allows us to work with square roots of negative numbers. Complex numbers are crucial because they provide solutions to certain equations that have no real solutions. For example, the equation \(x^2 + 1 = 0\) does not have a real solution, but it has a solution in terms of complex numbers, specifically \(x = i\) and \(x = -i\). Understanding complex numbers unlocks new possibilities in mathematical calculations and problem-solving, making them a fundamental tool in advanced mathematics.
To master complex numbers, practice rewriting expressions that involve imaginary numbers in their standard form and explore their applications in different mathematical contexts.

The Square Roots of Negative Numbers

Taking the square root of a negative number may seem impossible at first, but with the introduction of the imaginary unit \(i = \sqrt{-1}\), it becomes quite straightforward. This approach lets us express square roots of negative numbers in a manageable form for calculations. For example:

• The square root of \(-16\) is expressed as \(\sqrt{-16} = 4i\), since \(\sqrt{16} = 4\) and \(\sqrt{-1} = i\).
• Similarly, the square root of \(-49\) becomes \(\sqrt{-49} = 7i\), here \(\sqrt{49} = 7\).

By rewriting square roots of negative numbers using \(i\), we can easily transform difficult expressions into their complex number form. This method helps to simplify calculations, making it crucial to getting comfortable with the concept of \(i\). Practice this conversion regularly; it will greatly enhance your understanding and handling of square roots of negative numbers.

Combining Like Terms with Real and Imaginary Parts

Combining like terms is a fundamental skill in algebra, and it applies equally to expressions involving complex numbers. To simplify such expressions, separate terms into their real and imaginary components. Let's take the expression \(-2 + 4i + 7 - 7i\) and break it down:

• Combine the real terms: Identify and add or subtract the real number portions. For this expression, we have \(-2\) and \(+7\). Combining them gives \(5\).
• Combine the imaginary terms: Next, focus on the terms involving \(i\). In this expression, the imaginary parts are \(4i\) and \(-7i\). Combining them results in \(-3i\).

By organizing and combining like terms, the original expression simplifies to a neater form, \(5 - 3i\). Practice breaking down longer expressions into real and imaginary parts separately and combining like terms to strengthen your algebraic manipulation skills, especially with complex numbers.
This technique is very helpful in simplifying equations and solving problems involving complex numbers.