Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. \(\frac{6^{12}}{6^{9}}\)

Step-by-Step Solution

Answer

The expression is \(6^3\).

Step 1: Identify the Problem

We are given the expression \(\frac{6^{12}}{6^{9}}\) and asked to write it as a single base. This involves using the laws of exponents to combine the terms into one power of 6.

Step 2: Apply the Quotient Rule for Exponents

Recall the quotient rule for exponents: \(\frac{a^m}{a^n} = a^{m-n}\). This rule allows us to combine the exponents by subtraction when dividing like bases.

Step 3: Subtract the Exponents

Subtract the exponent in the denominator from the exponent in the numerator: \(12 - 9 = 3\). This means \(\frac{6^{12}}{6^{9}} = 6^{12-9} = 6^3\).

Step 4: Write the Final Expression

The expression \(6^3\) is a single base written with a positive exponent. This is the solution to our problem.

Key Concepts

Quotient Rule for Exponents

When you encounter a division of like bases with powers, the quotient rule for exponents is your key tool. The rule states that when you divide two exponents with the same base, you can subtract the exponent of the denominator from the exponent of the numerator. This can be expressed as \( \frac{a^m}{a^n} = a^{m-n} \). This makes calculations simpler by reducing complex expressions into a single term with just one base. This rule is particularly useful because it keeps you from performing laborious calculations. For instance, in the problem \( \frac{6^{12}}{6^{9}} \), by using the quotient rule, you can quickly find that it's equivalent to \( 6^{12-9} = 6^3 \) without needing to multiply long numbers, giving you a neat and concise result.

Simplifying Expressions

Simplifying expressions involves reducing them to a more basic or manageable form without changing the value. When it comes to exponents, simplification can make expressions easier to use, especially for further mathematical operations or solving equations. In expressions involving division of like bases, simplification is made more accessible by using the quotient rule. By transforming \( \frac{6^{12}}{6^{9}} \) into \( 6^3 \), for example, computations become more straightforward. Simplifying helps in focusing on the core values and understanding the initial problem better.Always keep in mind that simplification in exponents not only involves performing operations but also ensuring that the final form of the expression is as simple and understandable as possible, maintaining all terms with their reduced powers.

Positive Exponents

Working with positive exponents is crucial to making expressions accurate and applicable. A positive exponent indicates how many times a base is multiplied by itself, resulting in straightforward real numbers as outcomes. In the context of exponent rules, ensuring expressions remain with positive exponents is important since negative exponents often require additional steps to convert them into positive ones. From the exercise \( \frac{6^{12}}{6^{9}} \), all operations ensure that exponents remain positive, and the result \( 6^3 \) naturally holds a positive exponent.Keeping exponents positive leads to intuitive results, making it easier to understand and utilize them in further calculations. Positive exponents also simplify interactions in algebraic contexts, providing clean and direct interpretations of mathematical relationships.