Problem 19 - Algebra and Trigonometry

Question

me Evaluate the expression using \(x=3, y=4,\) and \(z=-1\). \(\sqrt{x^{2}+y^{2}}\)

Step-by-Step Solution

Answer

The evaluated expression is 5.

Step 1: Substitute the Values

First, substitute the given values of \(x = 3\), \(y = 4\), and \(z = -1\) into the expression. The expression \(\sqrt{x^2 + y^2}\) becomes \(\sqrt{3^2 + 4^2}\).

Step 2: Calculate the Squares

Calculate the squares of the substituted values: \(3^2 = 9\) and \(4^2 = 16\). The expression now becomes \(\sqrt{9 + 16}\).

Step 3: Add the Squares

Add the results of the squares obtained in Step 2: \(9 + 16 = 25\). The expression now is \(\sqrt{25}\).

Step 4: Calculate the Square Root

Find the square root of 25. \(\sqrt{25} = 5\). Thus, the evaluated expression is 5.

Key Concepts

Substitution

Substitution is a fundamental concept in algebra. It allows you to replace variables in an expression with actual numbers or other expressions. This step is crucial as it simplifies the calculation process by giving specific values to otherwise general variables. Consider the expression \( \sqrt{x^2 + y^2} \). If you substitute \( x = 3 \) and \( y = 4 \) into this expression, it transforms into \( \sqrt{3^2 + 4^2} \).

• Identify the variables you want to substitute.
• Replace each variable with its given value.
• This results in a more straightforward numerical expression.

Substitution is often used in algebra to test scenarios or solve problems where specific legal values of variables are needed. It is essential to perform substitutions accurately to ensure the results of further calculations are correct.

Square Root

The concept of a square root is fundamental in mathematics. The square root of a number \( n \) is a value that, when multiplied by itself, gives \( n \). The notation \( \sqrt{} \) indicates the operation of taking the square root.In the expression \( \sqrt{25} \), for example, the number 25 is the radicand, and the operation asks, "what number multiplied by itself gives 25?" The answer is 5.

• Ensure the number under the square root is a non-negative number for real number solutions.
• Find the square root using mental math, estimation, or a calculator if needed.

Square roots are pivotal in various mathematical calculations, especially in geometry and algebra, to simplify expressions or solve equations.

Expression Evaluation

Expression evaluation is the process of simplifying an algebraic expression to find a specific numerical value. This happens after substituting the values of variables, calculating powers such as squares, and performing arithmetic operations like addition or multiplication.For instance, once you've substituted \( x = 3 \) and \( y = 4 \) into \( \sqrt{x^2 + y^2} \), you calculate \( 3^2 = 9 \) and \( 4^2 = 16 \). You then add these results to get \( 9 + 16 = 25 \).

• Break down the expression step by step.
• Carry out mathematical operations in the correct order: powers first, then multiplication/division, and addition/subtraction last.
• Simplify the expression to find the final value.

Expression evaluation allows you to find concrete answers from abstract algebraic expressions, making it a critical skill in mathematics. The final result here, using these substitutions and operations, yields the value 5. This demonstrates how expressions can represent clear numerical realities once evaluated fully.