Problem 46 - College Algebra with Modeling and Visualization

Question

Exercises \(45-54:\) Use the quotient rule to simplify the expression. Use positive exponents to write your answer. $$ \frac{6^{2}}{6^{-7}} $$

Step-by-Step Solution

Answer

The simplified expression is \(6^9\).

Step 1: Identify the Quotient Rule

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). This rule applies when dividing like bases.

Step 2: Apply the Quotient Rule

Apply the quotient rule to the expression \( \frac{6^2}{6^{-7}} \). Here, \( m = 2 \) and \( n = -7 \). Substitute these into the quotient rule formula: \( a^{m-n} = 6^{2-(-7)} \).

Step 3: Simplify the Exponent

Calculate the exponent: \( 2 - (-7) = 2 + 7 = 9 \). Thus, the expression becomes \( 6^9 \).

Step 4: Express with Positive Exponents

The expression \( 6^9 \) already has a positive exponent, so no further changes are needed.

Key Concepts

Quotient rule

When handling expressions with exponents that involve division, the quotient rule offers a straightforward path to simplification. The rule is a special guideline within the realm of exponents, and it specifically applies to situations where you have the same base number being divided. In such cases, the quotient rule states that you subtract the exponent in the denominator from the exponent in the numerator. This rule can be expressed as:\[ \frac{a^m}{a^n} = a^{m-n} \]This means that if both numerator and denominator have the same base \(a\), and the exponents are \(m\) and \(n\), respectively, the simplified form will be \(a^{m-n}\). It’s an efficient rule that helps make complex expressions much simpler. In our exercise, applying the quotient rule to \(\frac{6^2}{6^{-7}}\) gives us the expression \(6^{2-(-7)}\), which simplifies the work significantly. Remember, it only works when the bases are identical.

Simplifying expressions

Simplifying expressions, especially those involving exponents, involves using certain rules and properties to make expressions shorter and easier to read. Here are some key steps when simplifying mathematical expressions involving division and exponents:

• Match the bases: Make sure the numbers being divided (numerator and denominator) have the same base.
• Apply the quotient rule: Subtract the exponent of the denominator from the exponent of the numerator.
• Recalculate the exponent: Do the arithmetic operation in the exponents to find the simplified form.

In our specific example: from \(6^{2}/6^{-7}\), by applying the quotient rule, we get \(6^{2 + 7} = 6^9\). It simplifies the expression to one power of six. Simplification doesn't stop there; it may involve other properties for different types of problems, like negative exponents or zero exponents, depending on the situation encountered.

Positive exponents

Positive exponents are a fundamental aspect of working with powers in mathematics, allowing numbers to be expressed in a straightforward and concise manner. An expression with a positive exponent, such as \(a^n\), indicates that the base \(a\) is multiplied by itself \(n\) times:\[ a^n = a \times a \times ... \times a \ (\text{n times}) \]The goal in many mathematical contexts, like our exercise, is to ensure the final answer uses positive exponents. Why?

• Clarity and ease of understanding since they are standard form.
• Positive exponents directly indicate the magnitude of repeated multiplication.

In the exercise discussed, \(6^9\) is already in terms of a positive exponent post-simplification. Positive exponents provide a clean and finalized form, especially when used to express solutions or final answers in many mathematical problems.