Problem 11 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$ 11^{3} \div 11^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{11} \).

Step 1: Understand the Power Division Rule

When dividing two expressions that have the same base, we subtract the exponents. The formula is \( a^m \div a^n = a^{m-n} \). Here, both expressions have the base 11.

Step 2: Apply the Rule to the Expression

Using the rule \( a^m \div a^n = a^{m-n} \), simplify the expression \( 11^3 \div 11^4 \).Subtract the exponents: \( 3 - 4 = -1 \).

Step 3: Write the Expression with Positive Exponents

After simplifying, we have \( 11^{-1} \). To write this with a positive exponent, use the rule \( a^{-b} = \frac{1}{a^b} \). Thus, \( 11^{-1} = \frac{1}{11} \).

Key Concepts

Power Division Rule

The Power Division Rule is a fundamental principle when dealing with exponents. It states that when you are dividing two numbers with the same base, you can subtract the exponent of the divisor from the exponent of the dividend. This rule can be expressed using the formula:

• \( a^m \div a^n = a^{m-n} \)

For instance, if both numbers have the base 11—just like in our problem—then you apply the Power Division Rule by subtracting the exponents.
To illustrate, let's take the expression \(11^3 \div 11^4\). Here, you will subtract 4 from 3, resulting in \(11^{-1}\). Understanding this rule is crucial because it forms the basis of simplifying many more complex algebraic expressions.

Negative Exponent

A negative exponent indicates that the base should be taken as the reciprocal. In simpler terms, if you see a negative exponent, such as \(a^{-b}\), it translates to \(\frac{1}{a^b}\). This means that instead of multiplying \(a\) by itself \(b\) times, you are instead taking the reciprocal of \(a\) raised to the \(b\)-th power.
For example, in our exercise, we end up with \(11^{-1}\) after applying the Power Division Rule. According to the negative exponent rule, this is equivalent to \(\frac{1}{11}\). Hence, the expression \(11^{-1}\) is simplified to \(\frac{1}{11}\) which is an expression with positive exponents. Negative exponents often appear in calculations involving powers and are a handy tool in algebraic manipulation.

Base 11 Exponents

Working with base 11 exponents involves the same principles as working with exponents of any other base, such as 10 or e.

• Exponents on base 11 follow similar rules regarding operations, including addition, subtraction, multiplication, and division of the exponents.
• In any problem involving such equations, your focus will generally be to simplify and to express results with positive exponents whenever possible.

In our exercise example, both numbers are expressed with base 11, \(11^3 \div 11^4\). Although the base is 11, the same exponent rules apply. Specifically, using the Power Division Rule and addressing negative exponents simplifies the expression to \(\frac{1}{11}\).
Base 11 is especially ubiquitous in areas like modular arithmetic and certain applications in computer science. Understanding how to manipulate exponents of any base lays the groundwork for advanced mathematical studies.