Problem 62 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ \frac{2^{3}-7}{5^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression following the order of operations: $$\frac{2^{3}-7}{5^{2}}$$ Answer: The simplified expression is $$\frac{1}{25}$$.

Step 1: Identify the Expression

The given expression to be simplified is: $$ \frac{2^{3}-7}{5^{2}} $$

Step 2: Solve the Exponents

The first operation we need to tackle in this expression is the exponent operation. Solve the exponents in both the numerator and the denominator: $$ \frac{2^{3}-7}{5^{2}} = \frac{8-7}{25} $$

Step 3: Perform Subtraction in the Numerator

We have one operation left in the numerator which is subtraction. Perform the subtraction: $$ \frac{8-7}{25} = \frac{1}{25} $$

Step 4: Simplified Expression

Now, all operations have been performed, and the given expression simplifies to the fraction: $$ \frac{1}{25} $$

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. This means that you multiply the base by itself as many times as the exponent indicates. For example, in the expression \(2^3\), the base is 2, and the exponent is 3, indicating that 2 should be multiplied by itself three times:

• \(2^3 = 2 \times 2 \times 2 = 8\)

In our original exercise, we applied exponentiation to both the numerator and the denominator. First, we calculated \(2^3\), which equals 8. Then, we calculated \(5^2\) in the denominator, which results in 25. By calculating these values, we could progress with simplifying the expression.

Subtraction

Subtraction is the arithmetic operation of finding the difference between two numbers. In the context of an algebraic expression, subtraction follows the order of operations, often occurring after exponentiation. Let's break down how subtraction was applied in the original problem.

Subtraction in the Numerator

In the expression \(\frac{8-7}{25}\), notice that after evaluating the exponent, we needed to perform a subtraction operation in the numerator.

• Here, we subtracted 7 from 8, resulting in 1.

This leaves us with the simplified form \(\frac{1}{25}\). Understanding subtraction within the order of operations ensures that expressions are evaluated correctly, giving an accurate result.

Numerator and Denominator

In a fraction, the numerator and denominator play a crucial role in determining its value. The numerator is the top part of a fraction that shows how many parts we are considering, while the denominator is the bottom part that indicates into how many equal parts the whole is divided.For our original exercise:

• Numerator: Initially \(2^3 - 7\), which simplifies to 1 after solving the exponent and performing subtraction.
• Denominator: Originally \(5^2\), which simplifies directly to 25 after solving the exponent.

Understanding these components individually can greatly ease the process of solving and simplifying algebraic fractions and ensures accuracy in your math calculations.

Algebraic Fractions

Algebraic fractions are fractions in which the numerator and/or the denominator are algebraic expressions. These fractions can include variables, numbers, or both and often require careful simplification using the order of operations.In the problem \(\frac{2^3-7}{5^2}\), we dealt with an algebraic fraction involving both integer constants and operations such as exponentiation and subtraction. Key points to consider when handling algebraic fractions:

• Always perform operations in the correct order: Parentheses, Exponents, Multiplication/Division (from left to right), Addition/Subtraction (from left to right).
• Simplify both the numerator and the denominator separately before combining them.

This process ensures that the fraction is fully simplified, as seen in the final expression \(\frac{1}{25}\). Keeping these key points in mind will allow you to handle more complex algebraic fraction problems effectively.