Problem 55 - Fundamentals of Mathematics

Question

(Section 3.5) Expand \(3^{7}\). Do not find the actual value.

Step-by-Step Solution

Answer

The expanded form is \(3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\).

Step 1: Understand Exponentiation

Exponentiation involves multiplying a base number by itself a certain number of times. In this case, the base is 3, and the exponent is 7, which means we multiply 3 by itself seven times.

Step 2: Write the Expression

To expand the given expression, we write out the multiplication that results from applying the exponent: \(3^7 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\).

Step 3: Recognize the Pattern

Each multiplication step maintains the same base (3), and each multiplication builds on the previous product. Expanding doesn't require solving; it's about expressing the repeated multiplication.

Key Concepts

Powers

When we talk about powers in mathematics, we are discussing how many times a certain number, known as the base, is multiplied by itself. This repeated multiplication can be expressed through an exponent, which is the tiny number found next to the base number. For example, in the expression \(3^7\), the base is 3, and the power, or exponent, is 7. This means 3 should be multiplied by itself a total of 7 times.
Understanding powers is essential as they serve as a shorthand notation to simplify and express what can be complex calculations. Instead of writing seven instances of multiplication, the power conveniently tells us exactly how many times the base number repeats. Simply put:

• The exponent indicates the number of times the base multiplies itself.
• Powers help in keeping mathematical expressions tidy and manageable.

Base number

The base number in a power expression serves as the foundation for exponential multiplication. It is the number that gets repeatedly multiplied, dictated by the exponent's value. In our example of expanding \(3^7\), the base is 3. This base number is central to the operation since each multiplication step is built upon this original figure.
The base number remains unchanged throughout the multiplication process. It is always the number you start with and continue to use as the foundation:

• The base is the number that appears in the repeated multiplications.
• A stable anchor in power expressions, the base doesn’t change even as the exponent increases.

Knowing how to identify and handle the base number is crucial in working with powers confidently.

Multiplication

Multiplication is a fundamental operation in mathematics that involves adding a number, the base in powers, to itself a specific number of times, defined by the exponent. In the context of powers, multiplication allows us to efficiently express repetitive addition. With powers like \(3^7\), this means multiplying 3 by itself seven times:
\[3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\]
The power of powers is maintaining control over what could otherwise be a long and complicated string of numbers. With multiplication, understanding this process in depth allows you to work consistently with larger numbers and longer expressions.

• Multiplication transforms addition into a quicker, scaled form.
• In powers, multiplication uses the base number repeatedly as determined by the exponent.

Grasping multiplication in this way helps to clarify why exponentiation is such a valuable tool in mathematics.