Problem 18 - College Algebra

Question

Evaluate each exponential expression. $$ \frac{3^{8}}{3^{4}} $$

Step-by-Step Solution

Answer

The result of simplifying the given expression is \(3^4\).

Step 1: Identify the base and the exponents

The problem \(\frac{3^{8}}{3^{4}}\) consists of two terms, both with base 3. The first term, the numerator, has an exponent of 8. The second term, the denominator, has an exponent of 4.

Step 2: Apply the quotient rule

Apply the rule \(a^n / a^m = a^{n-m} \) where a is the base and n and m are the exponents. In this case, a is 3, n is 8 and m is 4, giving us an expression 3^{8-4}

Step 3: Simplify the expression

Subtract the exponent 4 from 8 to get 4, which simplifies the expression to 3^4

Key Concepts

Quotient Rule

The quotient rule is a valuable tool when working with exponential expressions. When you have the same base in both the numerator and the denominator of a fraction, this rule helps you simplify the expression. The rule is expressed as follows: if you have

• \( a^n \) in the numerator
• \( a^m \) in the denominator

Then the quotient rule states that you can subtract the exponents if the bases are the same:\[\frac{a^n}{a^m} = a^{n-m}\]The calculation becomes much simpler when you apply this rule. You don't have to calculate the large exponential values first. This makes handling complex expressions less intimidating. Consider the expression \( \frac{3^8}{3^4} \): by applying the quotient rule, you simply subtract 4 from 8 to get 3 raised to the power of 4, or \( 3^4 \).

Simplifying Exponents

Simplifying exponents involves reducing the complexity of exponential expressions while still maintaining mathematical equivalence. When you have an expression like \( 3^4 \), it's already simple, but when you start with something like \( \frac{3^8}{3^4} \), you need to simplify it first. After applying the quotient rule, you perform the operation:

• Subtract the exponent in the denominator (4) from the exponent in the numerator (8)
• This gives you a new exponent of 4 on the common base 3

This results in the expression \( 3^4 \), which is much simpler and easier to handle. Simplifying exponents reduces the steps you might otherwise need to calculate bigger exponential numbers directly. In real-world scenarios, it allows for quicker computations and clearer results.

Base and Exponents

Understanding the components of an exponential expression is key to mastering algebra. Each expression consists of a base and an exponent. For example, in \( 3^8 \),

• The base is 3, the number you'll multiply by itself
• The exponent is 8, indicating how many times to multiply the base, i.e., \( 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)

These terms are critical. When you're working with expressions like \( \frac{3^8}{3^4} \), recognizing that both parts share the same base helps you apply rules like the quotient rule correctly. It’s crucial, especially when expressions get more complicated, to keep track of which number is the base and what powers they're raised to.

Algebraic Expressions

Algebraic expressions are a set of variables, constants, coefficients, and operators that together form an expression which can be evaluated. Exponential expressions are a part of algebraic expressions. When you're handling algebraic expressions like \( \frac{3^8}{3^4} \), you're often looking at manipulating numbers to find a simplified form. To evaluate these expressions:

• Identify all components (base and exponent)
• Apply algebraic principles, like the quotient rule, to simplify

Algebraic expressions can become quite complex with multiple terms and operations, but understanding the foundations such as base and exponent in exponential expressions makes solving them more manageable. Mastery of these skills lays the groundwork for solving more advanced algebraic operations and grasping more complex mathematical concepts.