Problem 58
Question
Simplify each expression. $$ (2+\sqrt{-1})+(-3+\sqrt{-16}) $$
Step-by-Step Solution
Verified Answer
\(-1 + 5i\)
1Step 1: Simplify the square roots of negative numbers
Recognize that \( \sqrt{-1} \) is the imaginary unit, denoted by \( i \), and \( \sqrt{-16} \) is \( 4i \) because \( \sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4i \).
2Step 2: Combine like terms
Combine the real numbers and the imaginary numbers separately: \((2 - 3) + (i + 4i)\).
3Step 3: Perform the addition
Add the real numbers together and the imaginary numbers together to get \(-1 + 5i\).
Key Concepts
Imaginary UnitCombine Like TermsSimplifying Square Roots
Imaginary Unit
In mathematics, the concept of an imaginary unit is crucial when dealing with complex numbers. The imaginary unit is defined as the square root of \textbf{-1} and is denoted by the symbol \textbf{i}. This enigmatic entity allows for the expression of numbers that cannot be represented on the traditional number line.
Understanding this concept is pivotal, especially when you encounter a square root of a negative number. For instance, in our exercise, the term \( \sqrt{-1} \) is simplified as i, and similarly \( \sqrt{-16} \) is simplified as \( 4i \) because \( \sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4 \times i \). This step unveils the way complex numbers are constructed using the imaginary unit and sets the stage for further simplification.
Understanding this concept is pivotal, especially when you encounter a square root of a negative number. For instance, in our exercise, the term \( \sqrt{-1} \) is simplified as i, and similarly \( \sqrt{-16} \) is simplified as \( 4i \) because \( \sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4 \times i \). This step unveils the way complex numbers are constructed using the imaginary unit and sets the stage for further simplification.
Combine Like Terms
When presented with an algebraic expression, the method to combine like terms is an important algebraic skill for simplification. Like terms are the components of an expression that contain the same variables raised to the same power, and potentially include the imaginary unit as well.
The technique of combining like terms involves adding or subtracting the coefficients of these like terms. In our exercise, we need to group and add the real numbers (2 and -3) and the imaginary numbers (i and 4i). This means we are in essence treating the imaginary unit as a common 'variable' and combining the terms accordingly. The expression \((2 - 3) + (i + 4i)\) shows this process; the real number terms combine to -1, and the imaginary terms combine to 5i, leading to a simplified expression.
The technique of combining like terms involves adding or subtracting the coefficients of these like terms. In our exercise, we need to group and add the real numbers (2 and -3) and the imaginary numbers (i and 4i). This means we are in essence treating the imaginary unit as a common 'variable' and combining the terms accordingly. The expression \((2 - 3) + (i + 4i)\) shows this process; the real number terms combine to -1, and the imaginary terms combine to 5i, leading to a simplified expression.
Simplifying Square Roots
The process of simplifying square roots involves finding an equivalent expression that has no square roots, radicals, or irrational numbers, when possible. However, when dealing with negative numbers under a square root, the result is an imaginary number, which is manageable with the imaginary unit, i.
In the given exercise, the term \( \sqrt{-16} \) was simplified by recognizing that \( \sqrt{16} \) is a perfect square, which equals 4. We then attached the imaginary unit i to it to indicate the square root of -1. Thus, simplifying square roots of negative numbers typically results in an integer multiplied by i. This approach to simplification helps keep complex numbers in a standard form, which is easier to work with and understand.
In the given exercise, the term \( \sqrt{-16} \) was simplified by recognizing that \( \sqrt{16} \) is a perfect square, which equals 4. We then attached the imaginary unit i to it to indicate the square root of -1. Thus, simplifying square roots of negative numbers typically results in an integer multiplied by i. This approach to simplification helps keep complex numbers in a standard form, which is easier to work with and understand.
Other exercises in this chapter
Problem 58
Without graphing, tell how many \(x\) -intercepts each function has. $$ y=0.25 x^{2}+2 x+4 $$
View solution Problem 58
Solve for \(x\) in terms of \(a\). $$ 3 x^{2}+a x=a^{2} $$
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Open-Ended Write a quadratic equation with the given solutions. 3 and 5
View solution Problem 58
Writing Describe the family of quadratic functions whose members each have \((3,4)\) as its vertex.
View solution