Problem 52 - Algebra A Combined Function

Question

Evaluate. $$ 7^{2}-3^{4} $$

Step-by-Step Solution

Answer

The evaluated expression is -32.

Step 1: Calculate 7 Squared

First, calculate the square of 7. When you square a number, you multiply the number by itself. Therefore, \( 7^2 = 7 \times 7 = 49 \).

Step 2: Calculate 3 to the Power of 4

Next, calculate \( 3^4 \). This means 3 multiplied by itself 4 times, which yields \( 3 \times 3 \times 3 \times 3 = 81 \).

Step 3: Subtract the Results

Now, subtract the result of \( 3^4 \) from \( 7^2 \). Therefore, the calculation is \( 49 - 81 = -32 \).

Key Concepts

Basic Arithmetic

Basic arithmetic is the foundation of all math operations, and it includes operations such as addition, subtraction, multiplication, and division. In this exercise, we specifically deal with exponentiation and subtraction.
Exponentiation is written as a number raised to the power of another number, such as in the expression \(7^2\) or \(3^4\).
This tells us to multiply the base number by itself as many times as indicated by the exponent. Here, we squared 7 and raised 3 to the power of 4.
Subtraction occurs after the powers are calculated, as seen in the equation, where the result of \(7^2\) is subtracted by \(3^4\).
Basic arithmetic operations often form the building blocks for more advanced topics in mathematics.
While the operations themselves are simple, applying them correctly according to math rules is crucial.

Integer Operations

Integer operations involve mathematical calculations using whole numbers, which include positive numbers, negative numbers, and zero. In the given exercise, both exponents and subtraction are used.
The problem results in integer values, since both \(49\) and \(81\) are integers.
When subtracting these values, your result may also be an integer, including negative integers as seen with the final result, \(-32\).

Performing integer operations requires attention to several key points:

• Recognize that exponents, like \(7^2\) and \(3^4\), result in positive integers when the base number is positive.
• When subtracting a larger number from a smaller one, the result will be a negative integer, as seen with \(49 - 81\).

Understanding integers and their operations is essential as they frequently appear in various forms across math problems.

Order of Operations

The order of operations is a set of rules used to determine the sequence in which calculations are performed.
This sequence is crucial to obtain a correct answer. These rules are often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

In this problem, we apply the order of operations by first performing calculations involving exponents:

• Calculate the exponent for each number separately (e.g., \(7^2\) and \(3^4\)).
• After resolving the exponents, subtraction is performed next.

Ignoring the order of operations or doing them in the wrong sequence can lead to incorrect results.
By following the correct order, you can solve complex expressions accurately. This becomes increasingly important as you encounter more advanced levels of mathematics.