Problem 70 - Algebra 1: Concepts and Skills

Question

Evaluate the radical expression when a = 2 and b = 4. $$ \sqrt{b^{2}-8 a} $$

Step-by-Step Solution

Answer

0

Step 1: Substitution

Substitute the given values into the expression. That is, replace a with 2 and b with 4. So, \( \sqrt{b^{2}-8 a} \) becomes \( \sqrt{4^{2}-8 * 2} \)

Step 2: Simplify Inside the Radical

Now it is necessary to simplify the expression inside the square root. This implies performing exponentiation and multiplication. The expression will be changed to \( \sqrt{16-16} \)

Step 3: Final Simplification

Continue with the simplification. The expression under the square root simplifies to 0. Taking the square root of 0 gives us the final answer, which is 0

Key Concepts

square root

The square root symbol is used to depict a number that, when multiplied by itself, yields a specified value. It is commonly denoted by the radical symbol \( \sqrt{} \). For example, the square root of 16, denoted by \( \sqrt{16} \), equals 4 because 4 times 4 equals 16.
In radical expressions like \( \sqrt{b^{2} - 8a} \), the number under the square root sign (the radicand) must be simplified first before taking the square root. It is crucial to ensure the value beneath the square root is non-negative prior to extracting its root to get a real number outcome.

substitution method

The substitution method in algebra is a technique used to evaluate expressions by replacing variables with given numerical values. This is particularly useful in problems involving radical expressions where specific values need to be substituted to simplify the expression.
In the given exercise, we substitute the values \( a = 2 \) and \( b = 4 \) into the expression \( \sqrt{b^{2} - 8a} \). The expression then changes from involving variables to a numeric calculation: \( \sqrt{4^{2} - 8 \times 2} \). This is the first step in evaluating the expression and is crucial for solving algebraic expressions involving variables.

simplification

Simplification refers to reducing an expression to its simplest form by performing arithmetic operations. For the expression \( \sqrt{b^{2} - 8a} \), you start by simplifying the expression inside the radical.
1. Calculate \( b^{2} \), which is \( 4^{2} = 16 \).
2. Compute \( 8 \times a \), which gives \( 8 \times 2 = 16 \).
3. Subtract the results: \( 16 - 16 = 0 \).
This simplification step is essential as it paves the way for the next step, which is finding the actual value of the square root of the simplified result.

expressions evaluation

Expressions evaluation is the process of determining the value of an expression once it has been fully simplified. After substituting the given values and simplifying the expression in the problem, you will evaluate the final expression: \( \sqrt{16 - 16} \).
The expression inside the square root has been simplified to 0, and the square root of 0 is simply 0.
This complete evaluation leads to the final result of the radical expression. Evaluating expressions is a critical step in mathematical problem-solving, ensuring that you derive the correct numerical answer from the expressions provided.