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Chapter 4: Problem 9

In Exercises 5-16, write the complex number in standard form. $$ \sqrt{-75} $$

Short Answer

Expert verified
The complex number in standard form is \(8.66i\).

Step by step solution

01

Identify the number under the square root

Identify that the number under the square root is -75. This is a negative number, so its square root will be an imaginary number.
02

Break down the number under the square root

-75 can be written as -1 * 75. The square root of -1 is \(i\) and the square root of 75 is a real number.
03

Find the square root of the real part

Find the square root of 75 which will be a real number. The square root of 75 is approximately 8.66 (rounded to two decimal places)
04

Combine the real and imaginary parts

Combine the real part from step 3 and the imaginary part from step 2 to get the imaginary number. We know \(i=\sqrt{-1}\), so the square root of -75 is \(8.66i\)

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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 intriguing part of mathematics. They come into play when we deal with the square roots of negative numbers. Normally, we can't find a real number whose square is negative. This is where imaginary numbers shine! Imaginary numbers are based on the mathematical symbol \(i\).
\(i\) is defined as \(\sqrt{-1}\). This simple yet powerful idea allows us to handle and calculate expressions involving the square roots of negative numbers. With \(i\), we can convert those negative square roots into imaginary numbers.
  • Any imaginary number can be written as \(bi\), where \(b\) is a real number.
  • For example, the imaginary number from \(\sqrt{-75}\) is \(8.66i\).
Remember, the magic lies in knowing that \(i\) represents \(\sqrt{-1}\). This single insight opens up many possibilities in complex number calculations.
Square Roots
When we hear "square root," we usually think of finding a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). But things get interesting with negative numbers! The square root of a negative number leads us into the world of imaginary numbers.
With \(\sqrt{-75}\), we need to separate the square root of the negative part from that of the positive part:
  • Break \(-75\) into \(-1 \times 75\).
  • The square root of \(-1\) is \(i\).
  • The square root of \(75\) is approximately \(8.66\).
Thus, \(\sqrt{-75} = 8.66i\). The introduction of \(i\) resolves the challenge of squaring a negative number, by keeping it in the realm of complex numbers.
Standard Form of a Complex Number
The standard form of a complex number allows us to neatly express numbers with both real and imaginary parts. This form is written as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.
For a complex number like \(8.66i\), think of it as \(0 + 8.66i\) to fit the standard form. Here:
  • \(a = 0\) as there is no real part.
  • \(b = 8.66\).
This form helps us consistently manage calculations and gives clarity. Whether adding or multiplying complex numbers, the standard form provides a reliable structure. Remembering \(a + bi\) makes working with complex numbers intuitive and straightforward.

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Most popular questions from this chapter

In Exercises 1-24, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ 2(\sqrt{3}+i)^{7} $$

In Exercises 65-74, use the Quadratic Formula to solve the quadratic equation. $$ x^{2}+6 x+10=0 $$

In Exercises 45-60, use the theorem on page 356 to find all the solutions of the equation and represent the solutions graphically. $$ x^{3}-27=0 $$

In Exercises 1-24, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ \left[2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)\right]^{6} $$

In Exercises 65-74, use the Quadratic Formula to solve the quadratic equation. $$ 4.5 x^{2}-3 x+12=0 $$
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