Problem 11 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

\(\frac{1}{11}\)

Step 1: Identify the Base and Exponents

The expression is \(11^3 \div 11^4\). Notice that both terms have the same base, which is 11. The exponents are 3 and 4, respectively.

Step 2: Apply the Quotient Rule for Exponents

When we divide expressions with the same base, we can subtract the exponents. Therefore, \(11^3 \div 11^4 = 11^{3-4}\).

Step 3: Simplify the Exponent

Subtract the exponents: \(3 - 4 = -1\). This gives us \(11^{-1}\).

Step 4: Convert to Positive Exponent

To express \(11^{-1}\) with a positive exponent, use the rule \(a^{-n} = \frac{1}{a^n}\). Therefore, \(11^{-1} = \frac{1}{11^1}\).

Step 5: Final Simplification

Simplify \(\frac{1}{11^1}\) to \(\frac{1}{11}\), since \(11^1 = 11\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical expressions into simpler, more understandable forms. This process helps in analyzing and solving mathematical problems effectively. In our exercise, the expression to be simplified is a division of two exponentials with the same base, namely, \(11^3 \div 11^4\). This seems complex at first glance, but with the right rules, it becomes straightforward.
When simplifying such expressions, the key is to look for factors or operations that can be reduced or cancelled out. Here, we notice that both terms have a common base, which hints at the possibility of using exponent rules, specifically the quotient rule. Simplifying expressions often leads to writing the expression in a more useful form that highlights essential properties, making subsequent mathematical operations easier to handle.

Positive Exponents

Positive exponents denote the number of times a base is multiplied by itself. Having a clear understanding of positive exponents is crucial because they are the standard form used in most mathematical writings. In our exercise, after applying the quotient rule, we initially obtained a negative exponent, \(11^{-1}\).
To convert this to a positive exponent, we use the rule \(a^{-n} = \frac{1}{a^n}\). This transformation is necessary because positive exponents are generally preferred to indicate clearly the nature of growth or division in mathematical expressions.

• \(a^n\) represents multiplication of the base \(a\) \(n\) number of times.
• Negative exponents are transformed to positive by "flipping" them to the denominator.

By applying this rule in our solution, \(11^{-1}\) becomes \(\frac{1}{11}\), aligning with the convention of expressing our final answer with positive exponents.

Quotient Rule for Exponents

The quotient rule is a fundamental concept in algebra used to simplify expressions where terms with the same base are divided. This rule states that \(\frac{a^m}{a^n}=a^{m-n}\), where \(a\) is the base, and \(m\) and \(n\) are the exponents. This simplifies calculations by reducing the exponent instead of multiplying or dividing large numbers directly.
In our case, we applied this rule to simplify \(11^3 \div 11^4\) by subtracting the exponents, resulting in \(11^{3-4} = 11^{-1}\). This approach saves time and simplifies the computational process. It highlights how recognizing the base and exponents at the onset can streamline the path to the solution.

• Ensure the bases are identical before applying the quotient rule.
• Simplify by subtracting the exponent of the divisor from the exponent of the dividend.

Understanding and effectively applying the quotient rule can demystify many algebraic simplification tasks, making it a valuable tool for students.