Question

Express each of the following products in exponential form: (a) \(3 \times 3 \times 3 \times 3\) (b) \(1 / 3 \times 1 / 3 \times 1 / 3 \times 1 / 3\)

Short answer

(a) \(3^4\); (b) \((\frac{1}{3})^4\).

Step-by-step solution

Identifying the Base and Frequency

In both parts of the problem, identify the common number that is being multiplied repeatedly. This common number is referred to as the base.

Counting the Number of Multiplications

Determine how many times the base is being multiplied by counting the number of terms in each product. This count becomes the exponent in the exponential expression.

Expressing in Exponential Form

For part (a), the common number is 3, which is being multiplied 4 times. Thus, the product is expressed as \(3^4\). For part (b), the common number is \(\frac{1}{3}\), being multiplied 4 times. Thus, the product is expressed as \((\frac{1}{3})^4\).

Key concepts

Identifying the Base

In exponential form, the base is the number or expression being multiplied by itself a certain number of times. It's a fundamental component that remains consistent throughout the multiplication process. Imagine a scenario where you are asked to multiply a number repeatedly. This number, no matter its value, becomes the base in exponential notation.
For example, if you see the product \(3 \times 3 \times 3 \times 3\), the base here is \(3\). It's the number that appears consistently in each multiplication. Similarly, in the product \(\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}\), the base is \(\frac{1}{3}\). Being able to identify the base accurately is crucial as it sets the stage for properly expressing an operation in exponential form.

Counting Exponents

Counting exponents is all about determining how many times you multiply the base by itself. This count gives you the exponent in the exponential expression.
Let's delve into the idea: Suppose you have \(3 \times 3 \times 3 \times 3\). Here, \(3\) (which is the base) is multiplied by itself four times. Thus, the exponent is \(4\), and this product can be represented as \(3^4\).
Similarly, for \(\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}\), \(\frac{1}{3}\) is multiplied by itself four times, hence it is expressed as \((\frac{1}{3})^4\). Each product term contributes to the total count of the exponents. Counting correctly ensures that you convey the product accurately in its exponential form.

Multiplying Fractions

Multiplying fractions follows the same basic idea as multiplying whole numbers, where the base is repeated a set number of times, but it also requires careful consideration of numerators and denominators.
When you multiply fractions like \(\frac{1}{3}\), observe this: each multiplication involves multiplying both the numerators and denominators. So, \(\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}\) is not simply \(3^4\), but rather, it involves \((1 \times 1 \times 1 \times 1)\) over \((3 \times 3 \times 3 \times 3)\).
The correct exponential form depicts this as \((\frac{1}{3})^4\) and represents the fraction's repeated self-multiplication. Understanding this distinction is key when dealing with exponential expressions involving fractions.