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Chapter 0: Problem 52

Find each square root. If it is not a real number, say so. $$ \sqrt{-121} $$

Short Answer

Expert verified
The square root of \(-121\) is \11i\.

Step by step solution

01

- Understand the problem

Determine whether the given number inside the square root is positive or negative.
02

- Recognize the nature of the number

Note that \(-121\) is a negative number.
03

- Evaluate the square root

Understand that the square root of a negative number is not a real number. The square root of \(-121\) can be expressed in terms of imaginary numbers.
04

- Express the square root in imaginary terms

Use the property \(\sqrt{-a} = i\sqrt{a}\) where \(i\) is the imaginary unit. So, \(\sqrt{-121} = i\sqrt{121} = 11i\).

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Key Concepts

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

negative numbers
Negative numbers are numbers that are less than zero. They are represented with a minus sign in front of them. Examples of negative numbers include a -3,-121, and -0.5.
Negative numbers are used in various real-life scenarios, such as debt, temperature below freezing, and altitudes below sea level.
When you take the square root of a negative number, it is important to remember that it doesn't yield a real number. This is because no real number squared will result in a negative number.
imaginary numbers
Imaginary numbers are numbers that when squared give a negative result. The imaginary unit is represented by the symbol 'i'. So, 'i' is defined such that i^2 = -1.
When dealing with square roots of negative numbers, we use this property to express the result in terms of i.
For example, the square root of -121 can be converted into an imaginary number using the formula \( \sqrt{-a} = i\sqrt{a} \).
Applying this property gives us \( \sqrt{-121} = i\sqrt{121} = 11i \).
Imaginary numbers are crucial in various fields such as engineering, physics, and complex number theory, making it easier to handle and solve problems involving negative square roots.
properties of square roots
Understanding the properties of square roots is essential for solving equations and simplifying expressions.
Here are some important properties to remember:
  • The square root of a product is the product of the square roots: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \)
  • The square root of a quotient is the quotient of the square roots: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)
  • When the number inside the square root is negative, the result will be an imaginary number:\( \sqrt{-a} = i\sqrt{a} \)
These properties can help break down complex problems into simpler parts, making it easier to find the solution.
Always remember that the square root function applied to a negative number takes you to the realm of imaginary numbers, which significantly expands your mathematical toolkit.

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Most popular questions from this chapter

Write each fraction as a decimal. For repeating decimals, write the answer by first using bar notation and then rounding to the nearest thousandth. See Example 11. $$ \frac{5}{6} $$

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Write each percent as a fraction. Give answers in lowest terms. See Example 13. $$ 51 \% $$

Determine whether each statement is true or false. The absolute value of any number is the same as the absolute value of its additive inverse.

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