Problem 74 - Algebra 1: Concepts and Skills

Question

Evaluate the radical expression when a = 2 and b = 4. $$ \frac{36-\sqrt{8 a}}{b} $$

Step-by-Step Solution

Answer

The evaluated expression is 8.

Step 1: Substitute the Given Values

Replace \(a\) with 2 and \(b\) with 4 in the expression: \( \frac{36-\sqrt{8 \cdot 2}}{4} \).

Step 2: Simplification (Inside the Square Root First)

Simplify the calculation inside the square root first, \(8 \cdot 2 = 16\), making the expression now to be \( \frac{36-\sqrt{16}}{4} \).

Step 3: Take Square Root

The square root of 16 is 4, so the expression becomes \( \frac{36- 4}{4} \).

Step 4: Execute Subtraction and Division

Subtract 4 from 36 to get 32, and divide by 4, which yields 8. So, the evaluated expression equals 8.

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics, especially when dealing with radical expressions. The square root of a number is another number that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because \(4 \times 4 = 16\). In our exercise, you first encounter the square root while simplifying \(\sqrt{8a}\). Square roots can sometimes result in neat whole numbers, as in our case with 16, but often they can be decimal or even irrational numbers. When evaluating square roots, it's important to ensure precision in calculations to achieve an accurate result.

Using Substitution in Expressions

Substitution is a handy technique for solving expressions. It involves replacing variables with specific values. In our exercise, the expression involves two variables, \(a\) and \(b\). By substituting \(a = 2\) and \(b = 4\), the expression \(\frac{36-\sqrt{8a}}{b}\) is transformed into \(\frac{36-\sqrt{8 \cdot 2}}{4}\). This step simplifies the process as we can now work with actual numbers rather than abstract variables. Substitution is crucial in evaluating expressions as it bridges the gap between theoretical expressions and practical calculations. It's like applying a recipe by adding specific ingredients to get the desired dish.

Steps to Simplification

Simplification is all about breaking down an expression into its simplest form. The goal is to make computations easier and more straightforward. In the exercise, once substitution is done, you simplify by addressing the operations within the square root first. Multiply 8 by 2 to obtain 16, making your expression \(\frac{36-\sqrt{16}}{4}\). Then, determine the square root of 16. By resolving operations in order, you avoid errors, simplify correctly, and efficiently reach the end result. Simplification is key in reducing clutter and potential calculation mishaps in algebraic expressions.

Subtleties of Division in Expressions

Division, especially in algebraic expressions, requires careful attention to both numerator and denominator. When you reach dividing portions of the expression, like \(\frac{32}{4}\), you should first ensure all prior operations are complete. In our case, after subtraction \(36 - 4 = 32\), the expression simplifies as you divide 32 by 4, yielding 8. Simplification of division in expressions helps in achieving the most reduced form of the answer. Always check your final result to ensure all calculations have been executed correctly, resulting in the correct final answer.