Problem 50 - Algebra 1

Question

Evaluate the expression. \(5 \cdot 5+3 \cdot 3 \cdot 3\)

Step-by-Step Solution

Answer

The result of the evaluated expression is 52.

Step 1: Apply exponentiation

\[3 \cdot 3 \cdot 3 = 27 \], this is the result of exponentiation as per the order of operation rules. The expression now becomes \(5 \cdot 5+27\).

Step 2: Apply multiplication

\[5 \cdot 5 = 25 \], according to order of operations multiplication should be applied next. So the modified expression is \(25+27\).

Step 3: Apply addition

\[25 + 27 = 52 \]; With multiplication and exponentiation performed, the addition operation is the last step which gives the final result.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to a power, which is an exponent. In this operation, the base is multiplied by itself as many times as the exponent indicates. For example, when you see an expression like \(3^3\), it means you multiply 3 by itself three times. So, \(3 \cdot 3 \cdot 3 = 27\).

Exponentiation is one of the first operations you perform when dealing with an expression that involves multiple operations. According to the order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), you handle any exponentiations before doing any multiplications, divisions, additions, or subtractions.

This ensures that the operations are carried out in a consistent and correct manner, giving us the accurate result from a complicated expression.

Multiplication

Multiplication is the process of repeated addition. If you have a multiplication like \(5 \cdot 5\), it means you add the number 5, five times: \(5 + 5 + 5 + 5 + 5\), resulting in 25.

In the order of operations, multiplication comes after any operations involving parentheses and exponents. Once any exponentiations have been performed, you move on to execute any multiplications and divisions, working from left to right in the expression.

Using the order of operations, ensures we solve expressions in a way that everyone understands because it maintains consistency, accuracy, and prevents errors. In our expression, after solving the exponentiation, we multiply \(5 \times 5\) to simplify our expression to \(25 + 27\).

Addition

Addition is fundamentally the process of combining two or more numbers to get a sum. When performed, it is the simplest arithmetic operation and is often the last to be done when evaluating expressions using the order of operations.

After dealing with any parentheses, exponents, multiplications, and divisions in an expression, you perform all additions and subtractions, moving from left to right.

For example, in our problem, after we've simplified through steps of exponentiation and multiplication, the last step is to add \(25\) to \(27\). Performing the addition, \(25 + 27 = 52\), gives the final answer for the expression. Understanding and applying the sequence of operations ensures that the expression is solved accurately, leading to the correct solution.