Problem 106 - Algebra 1

Question

EXPRESSIONS WITH FRACTION BARS Evaluate the expression. $$ \frac{(2+5)^{2}}{3^{2}-2} $$

Step-by-Step Solution

Answer

The simplified form of this expression is 7

Step 1: Solve the Parentheses

First, we need to solve the equation inside the parentheses which is \(2+5\). This equals to 7.

Step 2: Apply the Exponents

Applying the square to the result from step 1, \(7^{2}\) ,we get 49. Similarly, calculate the square of 3 which is \(3^{2}=9\).

Step 3: Final Calculation

Lastly, calculate the rest of the expression: divide 49 by the result of the subtraction \(9-2\) which equals 49/7 = 7

Key Concepts

Understanding Fraction Bars

Fraction bars are used to represent division between numbers or expressions. They are a crucial part of many algebraic expressions and often serve to organize information clearly. In the expression \(\frac{(2+5)^{2}}{3^{2}-2}\), the fraction bar separates the numerator, \((2+5)^{2}\), from the denominator, \(3^{2}-2\). This helps to ensure that each part of the expression is evaluated properly according to the order of operations.

When dealing with fraction bars, always solve the operations in the numerator and denominator before performing the division suggested by the fraction bar itself. The fraction bar acts like a giant division symbol and requires that we solve each side completely before dividing. By doing so, we avoid mistakes and ensure accurate results.

Mastering the Order of Operations

The order of operations is essential in solving mathematical expressions correctly. It's often remembered through the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our expression \(\frac{(2+5)^{2}}{3^{2}-2}\), following PEMDAS helps us break down and systematically solve the problem.

1. We start with the parentheses, solving \(2+5\) first because operations inside the parentheses must be completed before moving forward.2. Next, we address the exponents by squaring 7 obtained from the parentheses and squaring 3.3. Following that, perform any multiplication or division present in the problem—in this case, represented by the fraction bar.4. Finally, any addition or subtraction is carried out last, like the subtraction in the denominator, \(9-2\).

Using the order of operations ensures that we tackle expressions systematically and accurately.

Exploring Exponents

Exponents are a way of expressing repeated multiplication. They are written as a small number, known as the power, placed to the top right of a base number. For instance, in our expression, \(7^2\) means 7 multiplied by itself, which equals 49.

Exponents indicate how many times the base is used as a factor. When calculating adventitiously, take care to compute the exponent before any multiplication, division, addition, or subtraction is performed in the same level of parentheses. This ensures that the power is correctly applied, which can significantly impact the result.
In our example problem, also note the role of exponents in both the numerator and the denominator. Calculating \(3^2\) to get 9 is a separate step before any further arithmetic operations are performed in the denominator. This emphasizes the importance of respecting the specific operational order suggested by exponents when they appear in an expression.