Problem 39 - Elementary and Intermediate Algebra

Question

Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$ \frac{8^{12}}{8^{4}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 8^8 \).

Step 1: Identify the Base and Exponents

In the expression \( \frac{8^{12}}{8^{4}} \), both the numerator and denominator have the same base, which is 8. The numerator's exponent is 12, and the denominator's exponent is 4.

Step 2: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \) when \( a eq 0 \). This means you subtract the exponent in the denominator from the exponent in the numerator.

Step 3: Subtract the Exponents

Using the quotient rule, subtract the exponents: \( 12 - 4 = 8 \).

Step 4: Write the Simplified Expression

Using the result from the subtraction, the simplified expression is \( 8^8 \).

Key Concepts

Simplifying Expressions with Exponents

When dealing with exponents in mathematics, expressions might seem complex, but they can often be simplified using specific rules. Simplifying expressions with exponents often makes calculations easier and results clearer.

• If you encounter an expression like \( \frac{8^{12}}{8^4} \), you'll want to simplify it to reduce the complexity.
• The basic principle involves using exponent rules to rewrite and combine the terms.

A good starting point when simplifying is to take note of the base numbers and their corresponding exponents. In the original expression, the base number is identical (8 in this case), simplifying further.
Each simplification step relies on your understanding of exponent rules, which allow us to transform potentially lengthy expressions into more manageable forms.

Properties of Exponents

Understanding the properties of exponents is crucial when handling any expression with powers. These properties provide a set of rules that make simplifications straightforward.

• Product of Powers: When you multiply two powers of the same base, you add the exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers: Our current expression, \(\frac{8^{12}}{8^4}\), uses this rule. You subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Power: When taking a power of a power, you multiply the exponents, as in \((a^m)^n = a^{m \cdot n}\).

These rules offer a roadmap for transforming and simplifying expressions. They help in systematically reducing an expression down to its simplified form. Furthermore, these principles apply whether you're dealing with numbers or variables, making them versatile tools in algebra.

Algebraic Expressions

Algebraic expressions are the bread and butter of algebra. They consist of variables, numbers, and operations that conform to mathematical rules. An expression like \(\frac{8^{12}}{8^4}\) is considered algebraic, even though it only includes numbers, as it involves an operation—division—involving powers.

• Algebraic expressions can become quite complex, involving multiple terms and operations.
• Simplification helps in solving equations and capturing the essence of the given problem without unnecessary complexity.

Expressions can include not only integers but variables as well (e.g., \(\frac{x^{10}}{x^3}\)), and the same rules for exponents apply. The beauty of algebra lies in its ability to take these complex expressions and methodically break them down through simplification.
Understanding how to manipulate them is key to mastering algebra and solving equations efficiently.