Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(6^{5} \times 6^{-7}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{36}\).

Step 1: Recall the Laws of Exponents

The laws of exponents state that when multiplying like bases, we add the exponents: \(a^m \times a^n = a^{m+n}\). In this expression, both terms have the base \(6\).

Step 2: Apply the Law to the Given Expression

Using the law of exponents, apply it to the expression \(6^5 \times 6^{-7}\). Adding the exponents, we get: \(6^{5 + (-7)}\).

Step 3: Simplify the Exponent

Calculate the exponent by adding the numbers: \(5 + (-7) = -2\). Thus, the new expression is \(6^{-2}\).

Step 4: Convert to Positive Exponents

To express with positive exponents, recall that \(a^{-n} = \frac{1}{a^n}\). Therefore, \(6^{-2} = \frac{1}{6^2}\).

Step 5: Calculate the Final Answer

Calculate \(6^2\): \(6 \times 6 = 36\). Therefore, \(\frac{1}{6^2} = \frac{1}{36}\).

Key Concepts

Laws of Exponents

The laws of exponents form the foundation for working with exponential expressions. These rules help us handle operations involving powers in a systematic way.
The key laws include:

• Multiplication of Like Bases: When multiplying expressions with the same base, you simply add the exponents. For instance, if you have something like \( a^m \times a^n \), the result will be \( a^{m+n} \).
• Division of Like Bases: When dividing expressions with the same base, subtract the exponents, e.g., \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: When raising a power to another power, multiply the exponents, which gives you \( (a^m)^n = a^{m \times n} \).

These laws simplify expressions by reducing the complexity of computations. In our example, we used the law of multiplication of like bases to handle the expression \( 6^5 \times 6^{-7} \). By adding the exponents, we obtained \( 6^{5+(-7)} \), which is \( 6^{-2} \).

Simplifying Expressions

Simplifying expressions involving exponents aims to make them easier to understand and work with.
This process involves combining, reducing, or transforming parts of an expression using the laws of exponents.
Here’s how it works with our exercise:

• Combine Like Terms: When expressions have the same base, use the laws of exponents to either add or subtract the exponents.
• Reduce Negative Exponents: If you end up with a negative exponent, convert it to a positive exponent to simplify further.
• Simplification Result: The simplified form is often easier to work with in equations or when performing further operations.

For \( 6^5 \times 6^{-7} \), simplifying gives us \( 6^{-2} \), which is transformed into a positive exponent as \( \frac{1}{6^2} \). The final result, \( \frac{1}{36} \), is a more straightforward and clean representation.

Positive Exponents

Positive exponents represent simple multiplication of a number, whereas negative exponents indicate a reciprocal.
To fully understand why we convert negative exponents to positive exponents, consider these points:

• Negative Exponent as Reciprocal: The expression \( a^{-n} \) is equal to \( \frac{1}{a^n} \), which shifts the base from numerator to denominator.
• Preference for Positive Exponents: Mathematicians often prefer positive exponents because they are more intuitive and easier to interpret.

Therefore, when we transformed \( 6^{-2} \) to \( \frac{1}{6^2} \), it allowed us to calculate \( \frac{1}{36} \), making the expression simpler and clearer. This method ensures expressions with exponents are easily comparable and manageable.