Problem 45 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lessons 1.2,1.3) $$ 3^{5} $$

Step-by-Step Solution

Answer

The evaluated expression \(3^5\) equals 243.

Step 1: Understanding the Operation

Exponentiation is a mathematical operation in integers, real and complex numbers, and more. Written as \(b^e\), it consists of two parts, the base (\(b\)) and the exponent (\(e\)). Here, our base (\(b\)) is 3, and the exponent (\(e\)) is 5.

Step 2: Applying the Operation

We can expand the expression \(3^5\) as the base 3 being multiplied by itself 4 more times. It becomes: \(3 \times 3 \times 3 \times 3 \times 3\).

Step 3: Calculating the Result

The multiplication now needs to be performed. First, multiply the first two threes, yielding 9. Then, multiply the result by the third three, yielding 27. Multiply this result by the fourth three, getting 81. Finally, multiplying 81 by the last 3, we get 243.

Key Concepts

Evaluate Expressions

When we come across a term like 'evaluate expressions', it involves the process of performing mathematical operations to find the value of a given expression. In our example, the expression is the exponential notation \(3^{5}\), and to evaluate it means to calculate its actual numerical value. Evaluating expressions requires an understanding of mathematical rules and operations to simplify the expression step by step.

It's important to proceed methodically, breaking down the expression into understandable components and operations, and then to carry out those operations in the correct order. For instance, if the expression includes both exponents and multiplication, as it does in our example, we tackle the exponentiation first and carry on with the multiplication of the expanded terms.

Mathematical Operations

Mathematical operations refer to the basic processes we use to calculate numbers. These include addition, subtraction, multiplication, division, and exponentiation. Each operation has specific rules and properties that determine how it is applied when we solve mathematical problems.

Understanding these operations is essential for anyone studying mathematics as they form the foundation for more complex concepts. In the example of \(3^{5}\), we're focusing on multiple iterations of multiplication to understand exponentiation. While carrying out these operations, it's crucial to work systematically to avoid errors and to ensure that each step follows from the previous one accurately.

Powers and Exponents

Powers and exponents are a shorthand way of expressing repeated multiplication. When we use an exponent, we're saying that a number, known as the base, is being multiplied by itself a certain number of times indicated by the exponent. For instance, \(3^5\) tells us that the base 3 is multiplied by itself 4 additional times (since we count the initial instance of the base as well).

Breaking Down Exponentiation

In the expression \(3^{5}\), 3 is the base, and 5 is the exponent. This can be expanded to \(3 \times 3 \times 3 \times 3 \times 3\), making it clear that exponentiation is a form of shorthand to make writing and working with very large or very small numbers more convenient. Mastering the use of exponents is a key math skill that simplifies the way we work with large-scale calculations.