Question

Express each of the following products in exponential form: (a) \(2 \times 2 \times 2\) (b) \(1 / 2 \times 1 / 2 \times 1 / 2\)

Short answer

(a) \(2^3\); (b) \(\left(\frac{1}{2}\right)^3\).

Step-by-step solution

Analyze the Given Product

For part (a), identify the base number which is repeated in the multiplication. Here, the base number is 2, and it is multiplied by itself three times.

Convert to Exponential Form

Represent the multiplication of the same number (base) repeated multiple times using exponents. Since 2 is repeated three times, we express it as an exponent: \(2^3\).

Analyze the Second Product

For part (b), identify the base number again. The base in this case is \(\frac{1}{2}\), and it is also multiplied by itself three times.

Convert the Second Product to Exponential Form

Represent the multiplication of \(\frac{1}{2}\) repeated three times using exponents: \(\left(\frac{1}{2}\right)^3\). This indicates \(\frac{1}{2}\) is repeated three times in the multiplication.

Key concepts

Base Number

In mathematics, the concept of a "base number" is fundamental when dealing with exponential form. A base number is essentially the number that is being multiplied by itself when you are expressing a repeated multiplication pattern.
For instance, in the expression given in the task:

• For part (a), the base number is 2.
• For part (b), the base number is \(\frac{1}{2}\).

The base number is critical because it is the number you will see being raised to an exponent. Recognizing this can be the first step in easily converting long multiplication sequences into much more manageable exponential expressions. Understanding that the base stays constant while only the number of times it is multiplied changes, facilitates the transition from a perceived complex multiplication chain to a simple power of a number.

Multiplication

When you encounter repeated multiplication of the same number, it's important to understand how that translates into exponential form. Multiplication is the act of adding a number to itself a certain number of times, and this often leads to seeing the same number multiplied multiple times in a row.
Consider the examples:

• For part (a), the number 2 is multiplied by itself three times: \(2 \times 2 \times 2\).
• For part (b), the number \(\frac{1}{2}\) is similarly multiplied by itself three times: \(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\).

The essence of multiplication here is redundancy in action. It is a simple concept that when reiterated with the same base number, allows us to shift into exponential notation, which tidies up expressions and calculations. Understanding multiplication as repeated addition lays down the foundation for eventually uncovering exponential relationships.

Exponents

Exponents provide a concise way to express repeated multiplication of a base number. When you convert repeated multiplication of a number into "exponential form", you are leveraging the power of exponents.
An exponent tells you how many times to use the base number in a multiplication.
For example:

• In part (a), the base number 2 is used three times, so it is written exponent as \(2^3\).
• Similarly, for part (b), the fraction \(\frac{1}{2}\) is multiplied three times, which is expressed as \(\left(\frac{1}{2}\right)^3\).

The base number is what is being multiplied, and the exponent signals how many times this multiplication occurs. Exponents are a reflection of repeated multiplication and allow us to write these long multiplication statements in a more compact and often easier-to-manage notation. They are an integral part of simplifying expressions and are widely used in many areas of mathematics to denote large numbers efficiently.