Problem 64 - Algebra 1: Concepts and Skills

Question

Evaluate the radical expression when $a=-1 \text { and } b=5. $$ \sqrt{a^{2}+8} $$

Step-by-Step Solution

Answer

The result of evaluating the given radical expression is \(3\)

Step 1: Substitute the given value of a

Substitute \( a=-1 \) into the expression, we get the new expression as: \( \sqrt{(-1)^{2}+8} \)

Step 2: Simplify the expression

Now, simplify the expression to get: \( \sqrt{1^2+8} = \sqrt{1+8} = \sqrt{9} \).

Step 3: Compute the square root

Next, compute the square root of 9 to get the final answer as: \( \sqrt{9} = 3 \)

Key Concepts

Evaluating Expressions

Evaluating an expression involves figuring out the numerical value of a given algebraic expression by replacing variables with actual numbers. Consider the expression \( \sqrt{a^2 + 8} \). To evaluate the expression, you need to follow these simple steps:

• Identify the variables and the values you are given. In this case, we are given \( a = -1 \).
• Substitute these values into the expression.
• Perform the necessary arithmetic operations to simplify and get the final result.

By systematically plugging into variables and simplifying, you can efficiently evaluate expressions. This process turns an algebraic expression into a concrete numerical answer.

Substitution

Substitution is a fundamental concept, particularly when working with expressions involving variables. It involves replacing a variable with a specific numerical value, which is crucial in evaluating expressions correctly. Suppose you have the expression \( \sqrt{a^2 + 8} \) and you're asked to find its value when \( a = -1 \).

• First, locate the variable \( a \) in the expression.
• Replace \( a \) with \(-1\) to simplify the expression to \( \sqrt{(-1)^2 + 8} \).

By substituting the values accurately, you transform the variable expression into a potentially simpler problem, which sets the stage for simplification.

Simplifying Expressions

The process of simplifying expressions allows us to reduce an expression to its most basic form. Once substitution is complete, simplification is the next step.

• Look at the expression you derived after substitution: \( \sqrt{(-1)^2 + 8} \).
• Calculate each part: \((-1)^2\) results in \(1\), thus the expression becomes \(\sqrt{1 + 8}\).
• Continue by performing the addition to get \(\sqrt{9}\).
• Finally, find the square root of \(9\), which simplifies to \(3\). This means the expression evaluates to \(3\).

Simplifying an expression might seem complex at first, but by handling it piece by piece, it becomes a straightforward task. Always solve inside-out, dealing with the operations within the radical or parentheses first, to ensure accuracy throughout the process.