Problem 57 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ \left(3^{4}-4^{3}\right) \div 17 $$

Step-by-Step Solution

Answer

Question: Simplify the given expression according to the order of operations: (3^4 - 4^3) ÷ 17 Answer: 1

Step 1: Evaluate the Exponents

Evaluate the exponents inside the parentheses: $$ (3^{4} - 4^{3}) = (81 - 64) $$

Step 2: Perform the Operation inside Parentheses

Subtract the numbers inside the parentheses: $$ (81 - 64) = 17 $$

Step 3: Perform Division

Divide the result of the subtraction by 17: $$ \frac{17}{17} = 1 $$ Hence, the simplified expression is 1.

Key Concepts

Exponents

Exponents are a key concept that represent repeated multiplication of a number by itself. For example, in the expression \(3^4\), the base is 3, and the exponent is 4. This means you multiply 3 by itself four times. So, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
Exponents are an essential part of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Evaluating exponents should come after dealing with anything inside parentheses.
In our example, you first need to solve the exponents \(3^4\) and \(4^3\). You would compute \(3^4 = 81\) and \(4^3 = 64\) before moving on to other operations in the equation.

Parentheses

Parentheses in mathematical expressions are used to define the order in which operations are performed. According to the order of operations, you should always evaluate expressions inside parentheses first.
In our given problem \((3^4 - 4^3) \div 17\), after calculating the exponents inside the parentheses, the expression becomes \((81 - 64)\). We deal with the parentheses by performing the subtraction, resulting in 17. By doing this, you ensure the operation inside the parentheses is completed before any division or other operations.

• Focus on the expression inside parentheses.
• Evaluate each term separately before performing the arithmetic operation within.

This step ensures that the rest of the expression is correctly simplified according to standard mathematical conventions.

Division

Division is one of the basic operations and is performed after parentheses and exponents are resolved in PEMDAS. In the expression \((81 - 64) \div 17\), the division operation is the final step after simplifying what was inside the parentheses.
In our example, after determining that the result of the expression in the parentheses is 17, you then perform the division: \( \frac{17}{17} = 1 \).
Key points to keep in mind:

• After simplifying inside parentheses, resolve any remaining division as the final step.
• Make sure all preceding operations are correctly done before dividing.

Division simplifies the expression to its final form, helping you find the most reduced version of the expression.