Problem 34 - Elementary Algebra

Question

For the following problems, expand the quantities so that no exponents appear. $$ 4^{3} $$

Step-by-Step Solution

Answer

Question: Expand and simplify the expression \(4^{3}\). Answer: 64

Step 1: Understand the definition of exponentiation

The definition of exponentiation states that any base raised to an exponent means that we multiply the base by itself for as many times as the exponent indicates. For example, \(a^n = a \times a \times \ldots \times a\) (n times).

Step 2: Apply the definition of exponentiation to the problem

In the given problem, the base is 4 and the exponent is 3. Using the definition of exponentiation, we can rewrite the expression as: $$ 4^{3} = 4 \times 4 \times 4 $$

Step 3: Calculate the product

Now, we will multiply the base (4) by itself three times: $$ 4 \times 4 \times 4 = 16 \times 4 = 64 $$ The expanded form of the expression with no exponents is 64.

Key Concepts

Base and Exponent

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. The base is the number that is being multiplied. The exponent shows how many times the base is used as a factor. For example, in the expression \(4^3\), 4 is the base, and 3 is the exponent. This means that the base, 4, is multiplied by itself a total of three times.

• Base: The main number being multiplied, for example, 4 in \(4^3\).
• Exponent: The little number above the base telling us how many times to multiply the base by itself, for example, 3 in \(4^3\).

Understanding base and exponent helps in simplifying expressions and solving equations effectively. Becoming comfortable with this concept allows you to tackle a wide range of mathematical problems.

Expanding Expressions

Expanding an expression means to write it in a form where no exponents are visible. This process involves using the base and exponent to repeatedly multiply the base by itself until you expand fully. The expression \(4^3\) is actually shorthandfor multiplying 4 by itself three times: \(4 \times 4 \times 4\). By expanding an expression, you can more easily perform the next steps in calculations, like multiplication or addition, because you break it down into its simpler components.Here’s how you can expand:

• Start with the base and exponent, for example, \(4^3\).
• Rewrite it by multiplying the base by itself—a number of times equal to the exponent: \(4 \times 4 \times 4\).
• This helps visualize the operation and prepares you for the multiplication step.

Expanding expressions is a crucial step in simplifying equations and is often used in algebra to break down problems into easier tasks.

Multiplication

Once an expression is expanded, you perform multiplication to find the final value. In our example of \(4^3\), after expanding, we have \(4 \times 4 \times 4\). Now let's multiply

• First, multiply the first two numbers: \(4 \times 4 = 16\).
• Then, take the result (16) and multiply it by the next number in the sequence: \(16 \times 4 = 64\).
• The product, 64, is the outcome of multiplying all factors together.

Multiplication of expanded expressions translates the problem into a straightforward calculation. Each step in multiplying helps build confidence in handling larger and more complex equations. With practice, multiplication becomes an easy part of handling exponents and expressions.