Problem 107 - Algebra 1

Question

EXPRESSIONS WITH FRACTION BARS Evaluate the expression. $$ \frac{6+7^{2}}{3^{3}-9-7} $$

Step-by-Step Solution

Answer

The evaluated expression is 5

Step 1: Solve the Exponents

First, calculate the values where exponents are present. In our expression, we have \(7^{2}\) and \(3^{3}\). So, \(7^{2}\) = 49 and \(3^{3}\) = 27. After this calculation, our expression becomes: \(\frac{6+49}{27-9-7}\).

Step 2: Add and Subtract in Numerator and Denominator

Now, perform the addition and subtraction in the numerator and denominator respectively. Numerator becomes \(6 + 49\) which is 55 and denominator becomes \(27 - 9 - 7\) which is 11. So the expression now looks like this: \(\frac{55}{11}\).

Step 3: Divide the Numerator by the Denominator

Lastly, divide the numerator by the denominator to evaluate the fraction. Hence, \(55\div 11\)= 5. That's your final answer.

Key Concepts

Fraction Bar Evaluation

When evaluating expressions that involve fraction bars, like \[\frac{6+7^{2}}{3^{3}-9-7}\], it is crucial to understand the role of the fraction bar itself. The fraction bar acts as a division indicator and also groups the numerator and denominator parts of the expression. Hence, you must first resolve the operations in the numerator and the denominator separately before carrying out the division. Simplifying the parts above and below the fraction bar separately helps to avoid confusion, especially with more complex expressions. When working with the fraction bar, remember: resolve each part separately, then divide.

Exponent Rules

Understanding exponent rules is fundamental when evaluating expressions with exponents. An exponent, also called a power, tells us how many times to multiply a number by itself. For example, \(7^{2}\) means you multiply 7 by itself twice. Mastering the basic rules such as \(a^{n} \times a^{m} = a^{(n+m)}\), and \((a^{n})^{m} = a^{(n \times m)}\), among others, is essential. This step in the process dramatically simplifies the expression and prepares it for further operations. Always perform exponent calculations early on, as done in Step 1 of our example.

Order of Operations

To evaluate expressions correctly, one has to follow the order of operations, often remembered by the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). It is vital to apply these steps sequentially to avoid errors. Failing to follow the proper order can result in completely different outcomes. Our example demonstrates these rules with exponents calculated first (Step 1), followed by addition and subtraction (Step 2), and finally, division as the last step (Step 3).

Numerical Expression Simplification

Simplifying numerical expressions involves reducing them to their simplest form by carrying out operations and combining like terms where possible. Step 2 of the solution shows the simplification of the numerator and the denominator by performing addition and subtraction respectively. Once simplified, these parts are no longer cluttered with unnecessary terms, making it easier to understand and manage the numbers. After all operations are performed, as seen in Step 3, simplification results in the most reduced form of the expression, giving us the final answer.