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Chapter 2: Problem 66

Write the complex number in standard form. $$\sqrt{-5} \cdot \sqrt{-10}$$

Short Answer

Expert verified
The complex number in standard form is \(-5\sqrt{2}\).

Step by step solution

01

Compute the square roots as complex numbers

To start, compute the square roots of -5 and -10. The square root of -5 is \(i\sqrt{5}\), and the square root of -10 is \(i\sqrt{10}\)
02

Multiply the results

Multiply the two results to obtain: \(i\sqrt{5} * i\sqrt{10}\)
03

Simplify the multiplication

By multiplication, obtain: \(i^2\sqrt{50}\). We know \(i^2 = -1\). Thus, the multiplication simplifies to \(-\sqrt{50}\)
04

Simplify the result

The square root of 50 can be simplified to 5 times the square root of 2. Therefore, the resulting complex number in standard form becomes \(-5\sqrt{2}\).

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Key Concepts

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

Imaginary Numbers
Imaginary numbers are an extension of the number system that helps us work effectively with square roots of negative numbers. In the realm of mathematics, encountering the square roots of negative numbers gives us a special unit called the imaginary unit, denoted as \(i\).
  • By definition, \(i = \sqrt{-1}\). This allows us to express square roots of any negative number using \(i\).
  • For example, \(\sqrt{-5}\) becomes \(i\sqrt{5}\) and \(\sqrt{-10}\) turns into \(i\sqrt{10}\).
The significance of imaginary numbers is vast, particularly in fields like engineering and physics, where they provide valuable ways to solve complex equations. Imaginary numbers are always paired with real numbers to form complex numbers, allowing us to explore mathematics beyond the limitations of strictly real numbers.
Complex Multiplication
Complex multiplication involves multiplying two complex numbers and it follows specific properties that arise from the nature of imaginary numbers. When multiplying complex numbers:
  • We use the distributive property, treating \(i\) just like a variable.
  • Each part is multiplied separately, and the results are combined.
For \(\sqrt{-5} \cdot \sqrt{-10}\), the operation becomes \(i\sqrt{5} \cdot i\sqrt{10}\). Here are the steps:
  • Multiply \(i \times i\) to get \(i^2\).
  • \(i^2\) is equal to \(-1\). Therefore, \(i^2\sqrt{50}\) simplifies to \(-\sqrt{50}\).
This process shows us that multiplying these imaginary expressions adheres to rules similar to those in algebra, reinforcing the complex number system as a robust extension of our usual arithmetic.
Simplifying Radicals
Simplifying radicals is an essential skill when working with square roots, especially in the context of complex numbers. To simplify a radical expression:
  • Factor the number under the square root into its prime components.
  • Identify pairs of factors, since each pair can be taken out as a single base number.
Let's consider simplifying \(\sqrt{50}\):
  • The prime factorization of 50 is \(2 \cdot 5^2\).
  • The pair of 5s can be removed from under the square root, giving us \(5\sqrt{2}\).
In our example, we thus simplify \(-\sqrt{50}\) to \(-5\sqrt{2}\), resulting in the simplified complex number form for this particular problem. This approach reveals the elegant nature of mathematics in transforming complicate forms into simpler expressions.

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Most popular questions from this chapter

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-4 x^{3}+5 x^{2}-2 x-6\) (Hint: One factor is \(\left.x^{2}-2 x-2 .\right)\)

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-1$$

Sketch the graph of each polynomial function. Then count the number of real zeros of the function and the numbers of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? (a) \(f(x)=-x^{3}+9 x\) (b) \(f(x)=x^{4}-10 x^{2}+9\) (c) \(f(x)=x^{5}-16 x\)

A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{4}+6 x^{3}+10 x^{2}+6 x+9$$
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