Problem 103 - Intermediate Algebra

Question

Choose the correct letter or letters. No pencil is needed, just think your way through these. Which radical is not a real number? a. \(\sqrt{3}\) b. \(-\sqrt{11} \quad\) c. \(\sqrt[3]{-10} \quad\) d. \(\sqrt{-10}\)

Step-by-Step Solution

Answer

d. \( \sqrt{-10} \) is not a real number.

Step 1: Understand Real Numbers

Real numbers include both rational and irrational numbers, and they can be positive, negative, or zero.

Step 2: Radical and Real Numbers

A radical expression like \( \sqrt{a} \) or \( \sqrt[3]{a} \) is real if it results in a real number. Specifically, \( \sqrt{a} \) is not real if \( a < 0 \), because the square root of a negative number is not defined in the real number system.

Step 3: Evaluate Each Option

1. Option a: \( \sqrt{3} \) is a real number as 3 is positive. 2. Option b: \( -\sqrt{11} \) is a real number since it is just the negative of a real number (\( \sqrt{11} \)).3. Option c: \( \sqrt[3]{-10} \) is a real number because cube roots are defined for negative numbers via real roots.4. Option d: \( \sqrt{-10} \) is not a real number because you cannot take an even-root (square root) of a negative number under real numbers.

Key Concepts

Radical Expressions

Radical expressions involve the use of radicals, such as square roots and cube roots, to represent numbers. Radicals allow us to express numbers in different forms that reveal other properties, especially when dealing with real numbers. A radical expression commonly involves the radical sign \( \sqrt{} \) or \( \sqrt[n]{} \) where \( n \) is a positive integer. The index signifies which root you are taking, such as 2 for square roots and 3 for cube roots.

• Square root, \( \sqrt{a} \): This is the number that, when multiplied by itself (squared), gives the value \( a \).
• Cube root, \( \sqrt[3]{a} \): This is the number that, when multiplied by itself three times (cubed), results in \( a \).

Radical expressions are significant in mathematics because they simplify the expression and calculation of roots in various equations. Understanding how radicals work within the realm of real numbers is crucial, as certain radical expressions may not result in a real number.

Square Roots

Square roots are one of the most common radical expressions and involve finding the number which, when squared, results in the original number. Geometrically, the square root gives us the length of the side of a square with a given area. The notation for square roots is \( \sqrt{a} \), where \( a \) is the number we want to find the root of.
While square roots of positive numbers always yield a real number, the square root of a negative number does not exist within the realm of real numbers. Instead, these roots extend into the domain of complex numbers. For instance:

• \( \sqrt{16} = 4 \): This is easy because \( 4^2 = 16 \).
• \( \sqrt{-9} \): This is not a real number since no real number squared results in a negative number.
• Negative square roots like \( -\sqrt{11} \): These are real but simply the negative value of a square root, indicating a reflection of the positive root value along the number line.

Understanding square roots is vital to identifying which radical expressions result in real numbers and which get into complex territory.

Cube Roots

Cube roots serve as another type of radical expression and are particularly unique due to their behavior with negative numbers. The cube root of a number \( a \), denoted as \( \sqrt[3]{a} \), is a value that, when used three times in a multiplication (cubed), equals \( a \). Unlike square roots, cube roots of negative numbers do result in a real number. This is because a negative number times itself three times remains negative, preserving its real number status.
Let's look at examples:

• Cube root of a Positive Number: \( \sqrt[3]{27} = 3 \), because \( 3 \times 3 \times 3 = 27 \).
• Cube root of a Negative Number: \( \sqrt[3]{-8} = -2 \), as \( (-2) \times (-2) \times (-2) = -8 \).

Due to this property, cube roots are particularly useful in various mathematical and real-world applications where handling negative quantities is essential. Understanding cube roots broadens the scope of applying roots in solving problems involving both real and negative numbers.