Problem 7 - Algebra 1: Concepts and Skills

Question

Use the power of a power property to write the expression as a single power of the base. \(\left(2^{4}\right)^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(2^{12}\).

Step 1: Understand the problem

The problem is asking to simplify the expression \(\left(2^{4}\right)^{3}\) using the power of a power rule.

Step 2: Apply the power of a power rule

The power of a power rule states that \((a^m)^n\) equals to \(a^{mn}\). Apply this rule to the given expression. This gives \(2^{(4 \times 3)}\).

Step 3: Simplify the exponent

Perform the multiplication in the exponent. This gives \(2^{12}\).

Key Concepts

Power of a Power Property

When dealing with exponents, understanding the rules is essential. The "Power of a Power" property is an important rule in exponentiation. It simplifies expressions where a power is raised to another power. Let's dive into how this works.

This property tells us that when you have a base with an exponent that is itself raised to another power, you multiply the exponents together. If you have a base \(a\) with an exponent \(m\) which is then raised to another exponent \(n\), it can be simplified as:

• \((a^m)^n = a^{m \times n}\)

In our exercise, this property turns \((2^4)^3\) into \(2^{12}\). This simplification makes complex expressions easier to manage and understand. Remember, this property helps you reduce the number of steps in calculations and is widely used in algebra.

Simplifying Expressions

Simplifying expressions in mathematics is about reducing them to their simplest form. This often involves using algebraic rules, properties, and arithmetic operations.

With expressions involving exponents, like \((2^4)^3\), it's crucial to apply the appropriate rules correctly. Applying the "Power of a Power" property effortlessly changes complex expressions into manageable forms.
For example, in our case, simplifying \((2^4)^3\) to \(2^{12}\) involves these steps:

• Recognize the form: Understand what type of exponents you're dealing with.
• Apply the appropriate rule: Use the power of a power rule in this case.
• Conduct arithmetic operations: Perform the multiplication of exponents and arrive at \(2^{12}\).

Such simplifications facilitate further calculations or interpretations, making your mathematical work much more efficient.

Exponents Multiplication

Exponents multiplication often involves applying rules that help simplify expressions efficiently. When handling expressions with exponents, such as \((2^4)^3\), multiplying exponents is key.

The rule used here is simple: multiply the exponents when raising one power to another. This turns the process of simplifying:

• First recognize you have a power raised to another, \((2^4)^3\).
• Then, apply the rule: multiplying the exponents within the parentheses, \(4 \times 3\).
• Arrive at \(2^{12}\), where multiplication has simplified the expression.

In expressions involving multiple steps or larger numbers, mastering this multiplication can make complex calculations straightforward and save time.

Algebraic Rules

Algebraic rules form the backbone of simplifying mathematical expressions. In the context of our exercise \((2^4)^3\), several algebraic principles come into play, making it crucial to understand how these rules operate.

Key rules include:

• The power of a power rule: \((a^m)^n = a^{m \times n}\).
• The simplification of exponents to consolidate terms.

Applying such rules systematically not only simplifies calculations but also enhances your problem-solving speed. These rules help you recognize patterns and reduce complex expressions efficiently.

Moreover, proficient use of algebraic rules boosts overall confidence in handling a wide range of problems, from simple algebraic manipulations to more complicated calculations across different areas of mathematics.