Problem 24 - Fundamentals of Mathematics

Question

Expand the terms. (Do not find the actual value.) \(5^{3}\)

Step-by-Step Solution

Answer

Expand \(5^3\) as \(5 \times 5 \times 5\).

Step 1: Understand the Problem

We need to expand the expression \(5^3\), which means we express it as a product rather than calculating its value.

Step 2: Write the Exponent as a Product

The expression \(5^3\) can be expanded as a product of three factors of 5, which is \(5 \times 5 \times 5\).

Step 3: Confirm the Expansion

Ensure that the number of factors matches the exponent. Here, we have written 5 three times, as expected.

Key Concepts

Mathematical Expressions

In mathematics, expressions are used to represent numbers, operations, or a combination of both. **Mathematical expressions** serve as the foundation of many mathematical concepts and equations. They can include numbers, variables (like \(x\) or \(y\)), and operators such as addition, subtraction, multiplication, and division. Expressions can be of various forms, ranging from simple, like \(2 + 3\), to complex, like \(3x^2 + 2x - 5\). They are instrumental in conveying mathematical ideas in an easy-to-understand format without computing the exact outcome. When handling expressions, it’s essential to understand their role in solving problems, simplifying calculations, and modeling real-world situations. In the case of exponents, expressions represent repeated multiplication, contributing significantly to efficiency and clarity in mathematics.

Product of Factors

The idea of a **product of factors** is fundamental when dealing with exponents. Instead of lengthy multiplication, exponents offer a more concise notation. When given an expression like \(5^3\), you see the power notation which simplifies expressing a product of repeated factors.In plain terms, the expression \(5^3\) is a shorthand way to write the multiplication of the number 5 by itself three times. This expression expands to the mathematical form of \(5 \times 5 \times 5\). Key points to remember about the product of factors:

• The base (in this case, 5) is the number repeatedly multiplied.
• The exponent (3 in this example) tells us how many times the base is used as a factor.

Recognizing expressions as products of factors makes calculations more manageable and allows us to interpret mathematical problems more effectively.

Expansion of Terms

**Expansion of terms** refers to writing an expression so that each factor is explicitly shown, particularly useful in comprehending exponents. By expanding terms, we unfold complex expressions or powers into a more detailed, understandable product of factors.Consider the term \(5^3\). While this is a compact form of notation, expanding it involves expressing each multiplication step explicitly. It shows us exactly how many times the base 5 is used and allows us to see what's happening "inside" the exponent. Through expansion, learners can clearly visualize:

• The repeated multiplication process, which can often be a source of confusion for beginners.
• The relationship between the base and the exponent and how they interact numerically.

Being able to expand terms accurately boosts comprehension and immediately reveals the complexity of the expression without needing to compute actual values.