Problem 57 - Elementary Algebra Within Reach

Question

In Exercises 55-58, rewrite the expression in exponential form. $$ \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} $$

Step-by-Step Solution

Answer

The expression \(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\) in exponential form is \( (1/2)^6 \).

Step 1: Count the number of times the fraction is repeated

The expression consists of six fractions all equal to 1/2. This is the base for our exponent.

Step 2: Express the repeated multiplication in exponential form

Using the property of exponents, repeated multiplication of the same number (or fraction, in this case), can be expressed as that number raised to the total count of the repetitions. In this example, this would be \( (1/2)^6 \).

Key Concepts

Properties of Exponents

Exponents are powerful mathematical tools that let us express repeated multiplication easily and concisely. When we talk about the properties of exponents, we're discussing the rules governing how exponents work. These rules help us simplify expressions and solve problems related to powers. The essential properties include the product of powers, power of a power, and power of a product.

• Product of Powers: When you multiply the same base with different exponents, you add the exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another power, you multiply the exponents. For instance, \((a^m)^n = a^{m\cdot n}\).
• Power of a Product: When a product is raised to an exponent, each factor is raised to the exponent separately. For example, \((ab)^n = a^n \cdot b^n\).

Understanding these properties allows you to manipulate and simplify expressions, saving time and making calculations easier.

Repeated Multiplication

When we multiply a number by itself over and over again, this is known as repeated multiplication. For example, multiplying \( \frac{1}{2}\) by itself six times is a case of repeated multiplication. Instead of writing \(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \), we can express this in exponential form.
Exponential notation is a shortcut for writing repeated multiplication. It is expressed as a base number raised to an exponent, where the base is the number being multiplied and the exponent is the number of times it is multiplied.
This means the expression \(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\) can be rewritten as \(( \frac{1}{2})^6\). This tells us to multiply \(\frac{1}{2}\) by itself six times.

Fractions in Exponents

Exponents can also apply to fractions, not just whole numbers. When dealing with fractions as the base number in exponential form, we apply all the same rules of exponents. In the expression \((\frac{1}{2})^6\), the base is the fraction \(\frac{1}{2}\), and the exponent 6 tells us to multiply the fraction by itself six times.
To compute \((\frac{1}{2})^6\), you would carry out the multiplication step by step like this:

• First step: \(\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}\)
• Second step: \(\frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8}\)
• Continue this pattern until the sixth multiplication.
• The final result will be \(\frac{1}{64}\).

This systematic approach demonstrates how fractions in exponents require careful attention but are no different in principle from working with whole numbers. With practice, handling fractional exponents becomes a straightforward task.