Problem 40 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Then simplify the answer. $$ \frac{3^{3}+8-7}{2 \cdot 7} $$

Step-by-Step Solution

Answer

The simplified result of the expression is 2.

Step 1: Evaluate the Exponent

First compute the exponent \(3^{3}\), which is \(3*3*3 = 27\).

Step 2: Handle the Numerator

Use the result from the previous step and handle the operations in the numerator. So, \((27 + 8) - 7\) is calculated as \(35 - 7 = 28\).

Step 3: Perform Division

Now with the numerator computed, divide by the denominator, which is \(2 * 7 = 14\). So, \(28 / 14 = 2\)

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating expressions in algebra. It's like a set of traffic rules for mathematics, ensuring we all arrive at the same answer. To remember the correct sequence, many people use the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When evaluating the expression \( \frac{3^{3}+8-7}{2 \cdot 7} \), we first deal with the parentheses and exponents. That's why we compute \(3^{3}\) first, yielding 27. Following the order of operations, we then move on to tackle any multiplication or division before proceeding to addition and subtraction. Hence, after simplifying the numerator, the division by \(2 \cdot 7\) comes next. Skipping or rearranging these steps can lead to incorrect results, highlighting the importance of PEMDAS in algebra.

Exponential Notation

Exponential notation is a powerful tool in algebra that allows us to express repeated multiplication compactly. An expression such as \(3^{3}\), read as 'three to the power of three,' compactly represents \(3 \times 3 \times 3\).

It's important to understand that in exponential notation, the base number (3 in this case) is multiplied by itself as many times as the exponent (also called the power) indicates. Hence, \(3^{3}\) simplifies to 27 because we multiply 3 by itself 3 times. This neat notation helps greatly when working with large numbers or complex expressions since it makes calculations and the understanding of growth patterns easier.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill that involves reducing an expression to its simplest form, making it easier to understand and work with. As we simplify, we combine like terms, perform arithmetic operations, and apply properties of equality.

In our example \( \frac{3^{3}+8-7}{2 \cdot 7} \), after evaluating the exponent \(3^{3}\), we combine the numbers in the numerator, which are like terms since they are all constants. Then we divide the simplified numerator by the simplified denominator. The goal is to ensure that the expression is as concise as possible without changing its value. Simplification is like tidying up an equation, making sure it's neat and accessible for further analysis or problem solving.