Problem 107 - Intermediate Algebra

Question

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ 4^{1 / 2} $$

Step-by-Step Solution

Answer

The correct option is A.

Step 1: Rewrite the Expression

First, we rewrite the expression \(4^{1/2}\) to understand it better. The expression \(4^{1/2}\) means the square root of 4, which is written as \(\sqrt{4}\).

Step 2: Calculate the Square Root

Next, we compute the square root of 4. The square root of 4 is 2, as 2 multiplied by 2 equals 4 (i.e., \(2 \times 2 = 4\)).

Step 3: Match the Result to the Given Options

Finally, we match the result of the square root calculation to the options provided. Since \( \sqrt{4} = 2\), it corresponds to option \(A\).

Key Concepts

Square roots

The term "square root" refers to a value that, when multiplied by itself, gives the original number. Let's explore this using the example of the number 4.

• The square root of 4 is 2 because 2 multiplied by itself (2 \( \times \) 2) equals 4.
• It is represented mathematically as \( \sqrt{4} \).
• Notice the symbol "\( \sqrt{} \)" which denotes the square root.

Finding square roots is an essential skill in mathematics, as it is frequently used in equations and in taking measurements. By knowing how to calculate such roots, you can solve more complex problems and interpret mathematical expressions effectively.

Rational exponents

Rational exponents provide an alternative notation to express roots and powers simultaneously. They make mathematical expressions more compact and efficient to handle. Let's try to understand this with the expression \(4^{1/2}\).

• The exponent \(1/2\) indicates a square root operation. That's why \(4^{1/2}\) is equivalent to \(\sqrt{4}\).
• In general, an exponent like \(n/m\) translates to the \(m\)-th root of the base raised to the \(n\)-th power.
• This means \(4^{1/2} = 4^{0.5} = 2\).

Understanding rational exponents allows mathematicians to perform various algebraic operations easily, offering versatility in solving equations that involve roots.

Mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operations that define a specific value or equation. In our exercise, the expression \(4^{1/2}\) is a simple one, demonstrating how to represent roots using exponents.

• First, recognize that an expression like \(4^{1/2}\) encapsulates a mathematical concept—the square root transformation.
• Such expressions can include multiple operations and components, which, when combined, convey a specific mathematical statement or problem.
• Understanding these expressions is crucial for identifying solutions and manipulating mathematical equations logically.

By breaking down expressions into understandable components, you can decode complex problems and apply strategies that lead to a solution. This skill is invaluable in any area, from everyday calculations to advanced mathematical research.