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Chapter 2: Problem 10

In Exercises 1 to 10 , write the complex number in standard form. $$11+\sqrt{-48}$$

Short Answer

Expert verified
The complex number in standard form is \(11+4i \sqrt{3}\).

Step by step solution

01

Identify the square root of a negative number

In the given expression \(11+\sqrt{-48}\), we have the square root of a negative number. We can write \( \sqrt{-48} \) as \( \sqrt{-1*48} \)
02

Simplify the Square Root

The square root of -1 is 'i'. Replace \( \sqrt{-1}\) with \(i\). We have now \(i \sqrt{48}\). Now, simplify \(\sqrt{48}\), which can be written as \(\sqrt{16*3}\). The square root of 16 is 4, so we have \(4i \sqrt{3}\).
03

Write in Standard Form

Include this with the original real number in our expression for the final answer in a+bi form. The final solution is \(11+4i \sqrt{3}\).

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Key Concepts

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

Complex Numbers
When it comes to extending our understanding of numbers, complex numbers are a crucial addition. A complex number combines a real number with an imaginary number, leading to the standard form of a complex number: a + bi, where a and b are real numbers and i is the imaginary unit.

This form is practical because it allows for operations on complex numbers to be carried out similarly to those on polynomials. Complex numbers are used in various fields, such as engineering, physics, and computer science, where they provide solutions to equations that do not have real solutions.
Square Roots of Negative Numbers
The concept of square roots of negative numbers is initially perplexing since no real number multiplied by itself gives a negative result. However, in mathematics, to overcome this hurdle, the concept of imaginary numbers was introduced.

When we see a square root of a negative number, we can understand it as the product of the square root of -1, denoted as i, and the square root of the remaining positive number. For instance, \( \sqrt{-48} \) can be reinterpreted as \( \sqrt{-1} \sqrt{48} \) or i times \( \sqrt{48} \). This helps us handle complex calculations and adds depth to our number system.
Imaginary Unit
The imaginary unit, denoted by i, is defined as the square root of -1. It鈥檚 the foundation block of complex numbers and allows for the existence of numbers beyond the real number system.

The imaginary unit has a couple of essential properties, such as i2 = -1, which differentiates it from real numbers. This unit forms the 'imaginary' part of complex numbers and helps to express square roots of negative numbers within the system. Multiplying i by any real number transforms that real number into an imaginary number. This principle is the core tool used to handle operations involving the square roots of negative numbers.
Simplifying Square Roots
The process of simplifying square roots involves breaking down the radicand鈥攖he number under the square root symbol鈥攊nto its prime factors and removing any pairs of factors out from under the square root since the square root of a number's square is that number.

For example, with \( \sqrt{48} \), we break 48 into its prime factors, which are 2, 2, 2, 2, and 3. Grouping the pairs of 2s, we take two pairs out of the square root as 4, leaving us with \( 4 \sqrt{3} \), greatly simplifying the initial expression. This method allows for a clearer and often more useful representation in various mathematical contexts.

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Most popular questions from this chapter

Find a polynomial function \(P(x)\) with real coefficients that has the indicated zeros and satisfies the given conditions. Verify that \(P(x)=x^{3}-x^{2}-i x^{2}-9 x+9+9 i\) has \(1+i\) as a zero and that its conjugate \(1-i\) is not a zero. Explain why this does not contradict the Conjugate Pair Theorem.

Find a polynomial function of lowest degree with integer coefficients that has the given zeros. $$\frac{3}{4}, 2+7 i, 2-7 i$$

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=2 x^{3}+x^{2}-25 x+12$$

Evaluate \(\frac{x+4}{x^{2}-2 x-5}\) for \(x=-1\)

Find a polynomial function \(P(x)\) with real coefficients that has the indicated zeros and satisfies the given conditions. Verify that \(P(x)=x^{3}-x^{2}-i x^{2}-20 x+i x+20 i\) has a zero of \(i\) and that its conjugate \(-i\) is not a zero. Explain why this does not contradict the Conjugate Pair Theorem.
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