Problem 13 - Elementary Algebra

Question

For the following problems, expand each product (do not compute the actual value). \(2^{5}\)

Step-by-Step Solution

Answer

Question: Expand the product of \(2^{5}\) without computing the actual value. Answer: \(2^{5} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\)

Step 1: Understand the given expression

We are given the expression \(2^{5}\). It means that we will multiply 2 by itself 5 times.

Step 2: Expand the product

To expand the product, we will write down the multiplication of 2 repeated 5 times: \(2^{5} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) Now the expression is fully expanded, without calculating the actual value.

Key Concepts

Understanding Exponents

An exponent, also known as a 'power' or 'index', is a mathematical notation indicating the number of times a number (the base) is multiplied by itself. A simple example is seen in the expression 2^5, where 2 is the base and 5 is the exponent. This means 2 is multiplied by itself 5 times.

Exponents are not just limited to whole numbers. They can also be fractions, negative numbers, or even variables. The utility of exponents becomes evident when we need to represent very large or very small numbers efficiently, such as in scientific notation. Furthermore, understanding exponents is foundational for grasping more complex mathematical concepts like exponential growth, decay, and the calculations of interest rates in finance.

Exponents follow several fundamental laws that simplify computational efforts, such as the law of multiplying powers with the same base, which we will discuss further in the next section.

Multiplication of Powers

When multiplying powers with the same base, there's a simple rule to make things easier: you add the exponents together. The algebraic expression for this rule is a^m * a^n = a^(m+n). Let's consider an example to understand this concept better.

If you're multiplying 2^3 and 2^4, you don't need to expand each power and then multiply; you can simply add the exponents: 2^3 * 2^4 = 2^(3+4) = 2^7. This rule is very handy because it means you can perform operations on very large numbers without fully expanding them, which saves a lot of time and reduces the potential for errors.

This law of multiplication of powers is also crucial when working with algebraic expressions containing variables raised to exponents. It simplifies the process of multiplication and can be applied to variables as well as numerical bases.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division). Exponents are often a part of these expressions, adding a layer of complexity. A simple algebraic expression might look like 3x^2 + 4y - 5.

In algebra, expressions have to be manipulated according to the rules of algebra to solve equations or simplify expressions. It's like a puzzle where you have to find the value of the unknown variables. Expanding expressions, like the given 2^5, helps to understand the structure of algebraic expressions and prepares students for more complex operations such as factoring, solving polynomial equations, and dealing with rational expressions.

One of the key aspects of handling algebraic expressions is the ability to recognize like terms and combine them, a process that often relies on correctly managing exponents. By mastering both the concept of exponents and the multiplication of powers, you can handle a vast range of algebraic expressions with confidence.