Problem 67 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. \(6^{3} \cdot 6^{-1}\)

Step-by-Step Solution

Answer

The given expression simplifies and evaluates to 36.

Step 1: Apply the exponent rule

First observe that both terms are exponents of the same base \(6\). Therefore, using the rule of exponents \(a^{m} \cdot a^{n} = a^{m+n}\), we can simplify the given expression to \(6^{3-1}\).

Step 2: Simplify the exponent

In this step, compute the value of the exponent by subtracting 1 from 3, which gives \(6^{2}\).

Step 3: Evaluate the expression

Now, simply evaluate this into a single number by calculating the square of 6, which equals 36. Thus, \(6^{2} = 36\).

Key Concepts

Simplification of Expressions

Simplification is an essential skill in mathematics. It involves reducing expressions to their simplest form while maintaining their value. This allows you to see relationships and patterns more clearly. When working with expressions involving exponents, there are specific rules that help make simplification easier. These rules allow you to combine terms that share the same base or break them down into more manageable pieces.
Using the product of powers property, for example, allows you to combine like bases by adding their exponents: \(a^{m} \cdot a^{n} = a^{m+n}\).
This rule is powerful as it provides a direct way to simplify products of expressions with the same base, as demonstrated in our example expression \(6^{3} \cdot 6^{-1}\). By adding the exponents together \(3 + (-1)\), the expression simplifies to \(6^{2}\). Consequently, the simplification process transforms the original expression into a much simpler form, making it easier to evaluate.

Evaluating Exponents

Evaluating exponents is a process of calculating the power of a number. Essentially, it's finding the repeated multiplication of a number by itself, determined by the exponent. The exponent tells you how many times to multiply the base by itself. In \(6^2\), for instance, it means 6 is to be multiplied by itself 2 times: \(6 \cdot 6\).
Understanding evaluation is crucial as it connects the abstract numerical expression to a tangible value.

• The base is the number being multiplied.
• The exponent is the small number indicating how many times the base is multiplied.

After simplifying \(6^{3} \cdot 6^{-1}\) to \(6^2\), you move on to evaluate it to get 36. Breaking down each step, you can see how simplification aids in easily evaluating the expression.

Integer Bases

When dealing with exponents, integer bases are common and vital components of mathematical expressions. An integer base is a whole number that serves as the foundation for the exponential expression. It does not require any fractional or decimal part, thus making calculations quite straightforward.
For example, in the expression \(6^{3}\), the base is 6, which is an integer. Integer bases simplify various operations because they allow for direct multiplication without the complexity introduced by fractions or irrational numbers.

• Positive integer bases lead to positive results when raised to even powers.
• Negative integer bases flip sign depending on whether the exponent is even or odd.

Understanding how integer bases interact with their exponents is fundamental in working through exponential expressions, allowing you to easily manipulate and evaluate them correctly.