Problem 14 - College Algebra with Modeling and Visualization

Question

Simplify the expression without a calculator $$ \left(8^{27}\right)^{1 / 27} $$

Step-by-Step Solution

Answer

The simplified expression is 8.

Step 1: Understand the Expression

The expression given is \( \left(8^{27}\right)^{1 / 27} \). Here we have a power raised to another power. According to the laws of exponents, \((a^m)^n = a^{m \cdot n}\).

Step 2: Apply Exponent Simplification Rules

Using the rule from Step 1, we simplify the expression: \( (8^{27})^{1/27} = 8^{27 \times 1/27} \).

Step 3: Simplify the Exponent

Calculate the exponent by multiplying the exponents: \( 27 \times \frac{1}{27} = 1 \). This simplifies the power to \( 8^1 \).

Step 4: Final Simplification

With the exponent simplified to 1, the final expression becomes \( 8^1 = 8 \).

Key Concepts

Exponent Simplification

Exponent simplification is a fundamental concept when working with powers. It involves reducing expressions with exponents to their simplest form. One of the key rules for simplifying exponents is the power of a power rule, which states that when you have a power raised to another power, you can multiply the exponents. For example,

• If you have \( (a^m)^n \), you can simplify it to \( a^{m \cdot n} \).

In the original exercise, \( (8^{27})^{1/27} \) is simplified by multiplying the exponents \( 27 \) and \( 1/27 \), which result in an overall exponent of \( 1 \).
Exponent simplification helps to make complex expressions more manageable, so they are easier to evaluate or compare. The key is to always remember the core rules of exponents, which apply to all kinds of numbers—including fractions and negatives.

Power Rules

Understanding the power rules is essential for working with exponents. These rules are a set of guidelines that facilitate the manipulation and simplification of exponent-based expressions. They include:

• The Product of Powers Rule: \( a^m \times a^n = a^{m+n} \)
• The Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• The Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \)

These rules allow us to simplify mathematical expressions by combining and reducing exponents wherever possible. In particular, the power of a power rule is crucial in the original exercise, allowing us to address the expression \( (8^{27})^{1/27} \). By acknowledging these rules, we can transform complex expressions into simpler ones that are much more intuitive to work with.
Learning these power rules can simplify solving equations and improve computational efficiency, especially in more advanced algebra and calculus topics.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a value or series of operations. They are the building blocks of equations and are used extensively in mathematics to model scenarios and solve problems. When dealing with exponents in expressions, it's critical to understand how to manipulate and simplify them efficiently.
In the exercise given, \( (8^{27})^{1/27} \) is a mathematical expression featuring both a base and an exponent. The goal of the task is to simplify this expression using the laws of exponents, which allows us to rewrite it as \( 8^1 \) or simply \( 8 \).

• Expressions can become increasingly complex, involving multiple layers of operations and variables.
• The ability to simplify expressions like this is fundamental in reducing complex calculations to simple, actionable results.
• Mastering mathematical expressions is key to advancing in mathematics, as they form the core of problem-solving and analytical tasks.