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Chapter 0: Problem 1

Evaluate each expression or indicate that the root is not a real number. $$\sqrt{36}$$

Short Answer

Expert verified
The square root of 36 is 6.

Step by step solution

01

Understanding the Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we need to find a number that, if multiplied by itself, gives 36.
02

Evaluating the Square Root

The numbers that satisfy this condition (when squared, result in 36) are -6 and 6. However, by convention, when dealing with real numbers, the square root operation yields the positive root, unless otherwise specified. Therefore, the square root of 36 is 6 in the real number context.

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Key Concepts

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

Real Numbers
Real numbers are a vast and important set in mathematics. They include all the numbers that you can find on a number line, such as:
  • Positive numbers (like 1, 2, 3, ...)
  • Negative numbers (like -1, -2, -3, ...)
  • Zero
  • Fractions and decimals, like \(\frac{1}{2}\) or 0.75
One concept that does not belong to real numbers are imaginary numbers. Imaginary numbers arise when we take the square root of a negative number. For instance, the square root of -1 is represented as \(i\), an imaginary unit, which is not a real number.
In general, when dealing with square roots in the context of real numbers, it's essential to ensure that the number under the square root is non-negative. This is because the square root of a negative number does not exist in the realm of real numbers, making it an imaginary number.
Positive Root
When we talk about the square root of a number in the context of real numbers, we often refer to the principal or positive root. For example, when we find that \(\sqrt{36} = 6\), it implies the positive root.
Although a number can have both a positive and a negative root (because \(6^2 = 36\) and \((-6)^2 = 36\)), conventionally, the principal square root is considered to be the positive root. This helps maintain consistency in mathematical expressions and calculations.
Using the positive root is important for maintaining consistency and clarity in calculations:
  • It simplifies the mathematical process by focusing on one value instead of two.
  • It renders results consistent across different mathematical contexts and problems.
Evaluating Expressions
Evaluating mathematical expressions involves simplifying the expression to find its value. For expressions involving square roots, you need to determine the principal root unless otherwise specified.
Let's look at the example \(\sqrt{36}\). To evaluate this:
  • First, identify that you need a number which, when squared, results in 36.
  • The candidates here are 6 and -6, since both satisfy the condition \(x^2 = 36\).
  • Since we typically use the positive root in problem-solving, \(\sqrt{36}\) is evaluated as 6.
This evaluation is quick and follows from understanding both the properties of square roots and the rules about real numbers. It doesn't only apply to 36, but also to any other perfect square. Practicing with different numbers will help in mastering the art of evaluating expressions involving square roots.

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Most popular questions from this chapter

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and \(2012 .\) Also shown is the percentage of households in which a person of faith is married to someone with no religion. GRAPH CAN'T COPY. The formula $$I=\frac{1}{4} x+26$$ models the percentage of U.S. households with an interfaith marriage, \(I, x\) years after \(1988 .\) The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, \(N, x\) years after \(1988 .\) Use these models to solve Exercises \(107-108\). a. In which years will more than \(33 \%\) of U.S. households have an interfaith marriage? b. In which years will more than \(14 \%\) of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than \(33 \%\) of households have an interfaith marriage and more than \(14 \%\) have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than \(33 \%\) of households have an interfaith marriage or more than \(14 \%\) have a faith/no religion marriage?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

Find \(b\) such that \(\frac{7 x+4}{b}+13=x\) will have a solution set given by \(\\{-6\\}\)

Will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\)
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