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Chapter 10: Problem 36

Perform the operation and write the result in standard form. $$(11+3 i)(\sqrt{-25})$$

Short Answer

Expert verified
The result of the multiplication of the complex numbers \(11+3i\) and \(\sqrt{-25}\) in standard form is \(55i - 15\).

Step by step solution

01

Rewrite the square root as a complex number

We can write \(\sqrt{-25}\) as \(5i\). So the problem now is to multiply the complex numbers \(11+3i\) and \(5i\).
02

Multiply the complex numbers

Multiply \(11+3i\) and \(5i\) just like you would with binomial expressions by using the distributive property. We get \(55i + 15i^2\).
03

Simplify

Recall that \(i^2 = -1\). Substitute \(i^2\) in the expression, we get \(55i - 15\).

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Key Concepts

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

Standard Form
When working with complex numbers, expressing them in standard form is essential to simplify and organize our results. Complex numbers are generally written in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. For example, in the expression \(11 + 3i \), 11 is the real number while 3i is the imaginary component.
  • The standard form makes it easier to perform additions, subtractions, and even visualizations on the complex plane.
  • Keeping numbers in this format helps maintain consistency when comparing and simplifying complex expressions.
  • It clears ambiguity, as each part is explicitly marked as either real or imaginary.
In the original problem, after all operations are completed, it is crucial to rearrange the terms to get them into this standard form. This allows for a clear presentation and comprehension of the result, which is \(-15 + 55i\) in this case.
Imaginary Unit
The imaginary unit is a fundamental element of complex numbers. It is denoted by \(i\) and defined as the square root of \(-1\). This unique property is incredibly useful, as it allows us to work with square roots of negative numbers, which are not possible in the realm of real numbers. Here's how it works:
  • \(i^2\) equals \(-1\), which frequently appears in complex calculations.
  • When expressed in equations, the imaginary unit helps simplify expressions with roots of negative numbers.
  • For example, \(\sqrt{-25}\) converts to \(5i\), simplifying square root operations with negative under the root.
Understanding the imaginary unit is critical to performing operations on and simplifying complex numbers. In this problem, recognizing that \(\sqrt{-25} = 5i\) was a key step forward in solving it properly.
Distributive Property
The distributive property is a basic algebraic rule used extensively in both real and complex number calculations. It states that for any three numbers \(a\), \(b\), and \(c\), the expression \(a(b + c) = ab + ac\). This property is pivotal for expanding products of sums and was used in solving the given problem.
  • It allows multiplication over addition within an expression, simplifying calculations.
  • In the solution, when multiplying \((11 + 3i)\) and \(5i\), the distributive property was applied as follows: \(11 \times 5i + 3i \times 5i\).
  • After the application, the terms \(55i\) and \(15i^2\) emerged, allowing for further simplification of the expression.
The distributive property is not only applicable to real numbers but extends seamlessly to complex numbers, making it an indispensable tool in mathematics. By mastering it, students can more efficiently solve complex mathematical problems including those involving complex numbers.

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Most popular questions from this chapter

Testing Whether a Relation Is a Function In Exercises 7-12, determine whether the relation Input: Input: \(3,5,7 ;\) Output: \(d, e, f\) $$ \\{(3, d),(5, e),(7, f),(7, d)\\} $$

Evaluate the function as indicated, and simplify. \(h(s)=|s|+2\) (a) \(h(4)\) (b) \(h(-10)\) (c) \(h(-2)\) (d) \(h\left(\frac{3}{2}\right)\)

Repeated Reasoning When performing operations with numbers in \(i\)-form, you sometimes need to evaluate powers of the imaginary unit \(i\). The first eight powers of \(i\) are as follows. $$ \begin{aligned} &i^{1}=i \\ &i^{2}=-1 \\ &i^{3}=i\left(i^{2}\right)=i(-1)=-i \\ &i^{4}=\left(i^{2}\right)\left(i^{2}\right)=(-1)(-1)=1 \\ &i^{5}=i\left(i^{4}\right)=i(1)=i \\ &i^{6}=\left(i^{2}\right)\left(i^{4}\right)=(-1)(1)=-1 \\ &i^{7}=\left(i^{3}\right)\left(i^{4}\right)=(-i)(1)=-i \\ &i^{8}=\left(i^{4}\right)\left(i^{4}\right)=(1)(1)=1 \end{aligned} $$ Describe the pattern of the powers of \(i\).

Perform the operation(s) and write the result in standard form. $$(4+6 i)+(15+24 i)-10$$

Find the domain and range of the relation. $$\left\\{(2,16),(-9,-10),\left(\frac{1}{2}, 0\right)\right\\}$$
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