Problem 7 - Algebra and Trigonometry

Question

Evaluate each expression indicate that the root is not a real number. $$ \sqrt{25-16} $$

Step-by-Step Solution

Answer

The square root of \(25-16\) is 3, which is a real number.

Step 1: Perform Subtraction Inside the Square Root

Perform the subtraction operation inside the square root symbol. In this case, subtract 16 from 25, which yields 9.

Step 2: Evaluate the Square Root

Evaluate the square root of 9. The square root of 9 is 3, which is a real number.

Key Concepts

Square Root

Understanding the concept of square roots is crucial when evaluating mathematical expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by itself (3 x 3) equals 9.

• Symbol: The square root is represented by the symbol \( \sqrt{} \).
• Real vs Imaginary: A square root can result in a real number or an imaginary number. Real numbers are those you can find on the number line, whereas imaginary numbers arise when you take the square root of a negative number.

When working with expressions like \( \sqrt{25-16} \), it’s important to ensure that what’s under the square root (the radicand) is a non-negative number, ensuring the square root is a real number.

Subtraction

Subtraction is a fundamental arithmetic operation that involves taking one quantity away from another. It is often represented by the minus sign ('-'). Understanding how to perform subtraction accurately is essential for solving expressions.

• Working with Real Numbers: Real numbers include all the numbers on the number line, including both positive and negative numbers, as well as zero. Subtracting real numbers follows specific rules depending on whether the numbers are positive or negative.
• Order of Operations: In expressions, subtraction is performed according to the "PEMDAS/BODMAS" rule, which stands for Parentheses/Brackets, Exponents/Orders (including roots), Multiplication and Division, and Addition and Subtraction. Subtraction is often performed after any operations that occur within parentheses/brackets and any calculations of powers or roots.

For example, in the expression \( 25 - 16 \), you subtract 16 from 25, resulting in 9, which you then use for further calculations in the expression.

Expression Evaluation

Expression evaluation is the process of finding the value of a mathematical expression. It involves following the order of operations to compute the final result.To evaluate expressions like \( \sqrt{25-16} \):1. **Follow the Order**: Always begin from inside the parentheses or any grouping symbols, then move to exponents or roots.2. **Perform Arithmetic Operations**: Once inside the grouping symbols, perform subtraction or addition as necessary to simplify.

• Example: Begin with \( 25-16 \), compute to get 9.
• Square Root Evaluation: After simplifying inside, find the square root of the result. For \( \sqrt{9} \), the value is clearly 3, a real number.

Thus, effectively evaluating expressions requires both arithmetic skills and understanding of the order in which different operations should be applied.