Problem 25 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 5.

Step 1: Substitute Values into the Expression

To evaluate the given expression \( \sqrt{x^{2}+y^{2}} \), substitute \( x = 3 \) and \( y = 4 \) into the expression. This gives us: \[ \sqrt{3^{2} + 4^{2}} \].

Step 2: Calculate the Exponents

Calculate the squares of the numbers we substituted. \(3^2 = 9\) and \(4^2 = 16\). Therefore, the expression becomes \[ \sqrt{9 + 16} \].

Step 3: Compute the Sum Inside the Square Root

Add the results of the exponents calculated in the previous step: \(9 + 16 = 25\). Therefore, the expression is now \( \sqrt{25} \).

Step 4: Evaluate the Square Root

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

Key Concepts

Evaluating Expressions

Evaluating expressions involves working through math problems to find a numerical value, especially when variables are present. In our example, the expression is \( \sqrt{x^2 + y^2} \) and needs to be evaluated by substituting specific values for the variables given. This is done to transform a general expression into a specific, solvable problem. Let's break it down:

• Substitution: This means replacing variables like \(x\) and \(y\) with their respective values, here \(3\) and \(4\).
• Order of operations: Remember to handle operations such as exponents and square roots in the correct sequence.

Evaluating expressions accurately is an essential skill in algebra that comes in handy in various real-world settings.

Square Roots

A square root refers to a number that, when multiplied by itself, results in the original number. In notation, the square root of a number \(a\) is represented as \(\sqrt{a}\). In our problem, the evaluation process ends with determining \(\sqrt{25}\). Here's how it all works:

• Perfect Squares: A number like \(25\) is a perfect square because it can be expressed as \(5^2 = 25\).
• Common Square Roots: Knowing the square roots of common perfect squares (like \(4, 9, 16, 25\), etc.) helps in quickly evaluating expressions.

Finding square roots is a common task in algebra and can be beautifully simple if you remember perfect squares and their roots.

Substitution Method

The substitution method is a crucial aspect of solving algebra problems by replacing variables with their known values. This approach simplifies expressions and equations, making them easier to solve. Here's how substitution plays out in our case:

• Identifying Variables: Recognize which variables in the expression need replacement. Here, they are \(x\) and \(y\).
• Replacing with given values: Substitute \(3\) for \(x\) and \(4\) for \(y\) in the expression \(\sqrt{x^2 + y^2}\), resulting in \(\sqrt{3^2 + 4^2}\).

By simplifying an expression via substitution, you make it possible to solve it step by step, ensuring you focus on one part at a time, leading to accurate solutions.