Problem 57 - Algebra 1: Concepts and Skills

Question

Evaluate the expression \(\frac{3^{2}+6-5}{2 \cdot 5}.\) $$(F)1\quad (G)5\quad\quad (H)7\quad (J)10$$

Step-by-Step Solution

Answer

The evaluated expression is 1. Therefore, the correct answer is (F)1.

Step 1: Apply Order of Operations - Exponent

First, calculate the exponent. Here, it is \(3^{2}\), which equals 9. Thus, the expression becomes \(\frac{9+6-5}{2 \cdot 5}\).

Step 2: Apply Order of Operations - Addition and Subtraction

Next, perform the addition and the subtraction in the numerator. Therefore, the expression becomes \(\frac{9+1}{2 \cdot 5} = \frac{10}{2 \cdot 5}\).

Step 3: Apply Order of Operations - Multiplication and Division

Lastly, perform the multiplication in the denominator, and then do the division. The expression thus simplifies to \(\frac{10}{10} = 1\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating algebraic expressions. It's like reading a recipe; missing a step can lead to an entirely different outcome. In algebra, the standard order is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). To remember this sequence, you might have heard of the acronym PEMDAS or the phrase 'Please Excuse My Dear Aunt Sally.'

For example, given the expression \(\frac{3^{2}+6-5}{2 \cdot 5}\), you'll apply these rules step by step. The exponent \(3^{2}\) comes first before tackling any addition, subtraction, or multiplication. Without following this precise order, you could end up with an incorrect answer. Once the exponent is calculated, add and subtract in the numerator, then finally address any multiplication or division in the denominator.

Exponentiation in Algebra

Exponentiation is a form of mathematical shorthand that tells us how many times to multiply a number by itself. In algebra, an exponent applies only to the number or variable it's directly connected to. For instance, in the expression \(3^{2}\), the number 3 is the base and 2 is the exponent, which means you'd multiply 3 by itself twice: \(3 \times 3 = 9\).

This operation comes before any others when you're working within the order of operations. But be careful not to rush—ensure you’ve fully simplified any exponents before moving on to multiplication, division, or addition and subtraction! When done correctly, exponentiation can dramatically change the value of an algebraic expression, as seen in our original exercise.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that show the value of something. You can think of them as sentences in the language of mathematics where the value of the whole expression can change depending on the values of its individual parts.

When you work with expressions, disciplining yourself to follow the order of operations without skipping steps is essential to obtain the correct solution. Failure to properly evaluate exponentiation, for example, can lead you to a completely different answer. It is important to also consider the context; you could be finding the area of a square with side length \(3^{2}\), or you might be calculating interest compounded annually. Always read the entire expression, like \(\frac{3^{2}+6-5}{2 \cdot 5}\), carefully to determine which operations apply and in which order, just as you would read a full sentence to understand its meaning.