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Chapter 1: Problem 23

Write each expression in terms of \(i\) and simplify. (See Examples \(1-2)\) $$-\sqrt{-16}$$

Short Answer

Expert verified
-4i

Step by step solution

01

Recognize the Negative Square Root

Identify that the square root has a negative number inside. Since \(-16\) is negative, you need to involve the imaginary unit \(i\), where \(i = \sqrt{-1}\).
02

Factor the Radicand

Rewrite \(-16\) as \(16 \text{times} -1\). This gives us \(-\sqrt{-16} = -\big(\text{sqrt}(-1 \times 16)\big)\).
03

Simplify Using \(i\)

Use the property of square roots to separate the terms: \(-\big(\text{sqrt}(-1) \times \sqrt{16}\big) = -\big(i \times 4\big)\).
04

Simplify the Expression

Multiply the constants to get the simplified form: \(-\big(4i\big) = -4i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complex numbers
Complex numbers are numbers that have both a real part and an imaginary part. The imaginary unit is represented by the letter \( i \), which is defined as \( \sqrt{-1} \). The form of a complex number is usually written as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. For example, in the expression \( 3 + 4i \), 3 is the real part and 4i is the imaginary part.
  • \( a \) is the real part
  • \( bi \) is the imaginary part

Key Points:
  • The real part of a complex number can be any real number.
  • The imaginary part is a real number multiplied by the imaginary unit \( i \).
  • Imaginary numbers are a crucial part of complex numbers.
negative square roots
Negative square roots involve taking the square root of a negative number. Normally, square roots of negative numbers don't exist within the set of real numbers. To handle these, we introduce the imaginary unit \( i \), where \( i = \sqrt{-1} \). This allows us to simplify expressions that involve negative square roots.

How to Handle Negative Square Roots:
  • Identify the negative inside the square root.
  • Use \( i \) to represent the square root of \( -1 \).
  • Rewrite the negative number as a product of a positive number and \( -1 \).
For example, in \( \sqrt{-16} \), we can rewrite this as \( \sqrt{16 \times -1} \). Using \( i \), this becomes \( 4i \), because \( \sqrt{16} = 4 \) and \( \sqrt{-1} = i \). This demonstrates how negative square roots are simplified using imaginary numbers.
simplifying expressions
Simplifying expressions involves breaking down complex mathematical expressions into simpler, more manageable forms. For expressions involving imaginary numbers, follow these steps:

Steps to Simplify:
  • Identify and factor the radicand (the number inside the square root).
  • Separate the square root of any negative number using \( i \).
  • Combine like terms if possible.
This is what we do in the given problem:
1. Recognize \( -\sqrt{-16} \).
2. Rewrite \( -16 \) as \( 16 \times -1 \).
3. Use \( i \) to separate the terms: \( \sqrt{-16} = \sqrt{16 \times -1} \).
4. We get \( 4i \) because \( \sqrt{16} = 4 \) and \( \sqrt{-1} = i \).
5. Finally, the expression simplifies to \( -4i \).
Understanding these steps will help you easily simplify complex expressions, especially those involving negative square roots and imaginary numbers.

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Most popular questions from this chapter

A police officer uses a radar detector to determine that a motorist is traveling \(34 \mathrm{mph}\) in a \(25 \mathrm{mph}\) school zone. The driver goes to court and argues that the radar detector is not accurate. The manufacturer claims that the radar detector is calibrated to be in error by no more than 3 mph. a. If \(x\) represents the motorist's actual speed, write an inequality that represents an interval in which to estimate \(x\). b. Solve the inequality and interpret the answer. Should the motorist receive a ticket?

Write each phrase as an algebraic expression. Use \(x\) as the variable unless otherwise indicated. If \(x\) represents the larger of two consecutive odd integers, write an expression for the next smaller consecutive odd integer.

Determine if the statement is true or false. a. \(\mathbb{R} \subset \mathbb{C}\) b. \(\mathbb{C} \subset \mathbb{R}\)

Evaluate \(\sqrt{b^{2}-4 a c}\) for the given values of \(a, b,\) and \(c,\) and simplify. $$a=2, b=-6, \text { and } c=5$$

Consider a seesaw with two children of masses \(m_{1}\) and \(m_{2}\) on either side. Suppose that the position of the fulcrum (pivot point) is labeled as the origin, \(x=0 .\) Further suppose that the position of each child relative to the origin is \(x_{1}\) and \(x_{2}\), respectively. The seesaw will be in equilibrium if \(m_{1} x_{1}+m_{2} x_{2}=0 .\) Use this equation. Find \(x_{1}\) so that the system of masses is in equilibrium. \(m_{1}=64 \mathrm{~kg}, x_{1}=?\) and \(m_{2}=80 \mathrm{~kg}, x_{2}=2 \mathrm{~m}\)
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