Problem 7 - College Algebra with Modeling and Visualization

Question

$$ \left(2^{m}\right)^{2}=______ $$

Step-by-Step Solution

Answer

\((2^m)^2 = 2^{2m}\)

Step 1: Understand the Base and the Exponent

Identify that the expression \((2^m)^2\) follows the rule of exponents where a power is raised to another power.

Step 2: Apply the Exponentiation Rule

According to the power of a power rule, \((a^m)^n = a^{m imes n}\). Here, this translates to \((2^m)^2 = 2^{m \times 2}\).

Step 3: Simplify the Expression

Write the simplified expression by multiplying the exponents: \(m \times 2 = 2m\). Thus, \(2^{m \times 2} = 2^{2m}\).

Key Concepts

Power of a Power Rule

When you encounter an expression where an exponent is raised to another power, like in \((a^m)^n\), you're dealing with the power of a power rule. This rule is an essential part of understanding exponents. It helps simplify expressions by combining exponents.Here's how it works:

• The first exponent, \(m\), is being raised to yet another exponent, \(n\).
• According to the power of a power rule, the resulting exponent is the product of these two: \((a^m)^n = a^{m \times n}\).

This means you multiply the exponents. It turns a potentially complex expression into a single, simpler term.For example, in the original problem \((2^m)^2\), you use this rule to multiply the exponents: \(m \times 2\) simplifies to \(2m\), giving the result \(2^{2m}\). It's a shortcut that avoids expanding the powers separately.

Exponents

Exponents are a mathematical way to express repeated multiplication of the same number. They appear frequently in algebra and several other areas of mathematics.Here are a few basics about exponents:

• An exponent represents how many times a number, called the base, is multiplied by itself.
• In an expression like \(2^m\), \(2\) serves as the base and \(m\) is the exponent.

Exponents make it much easier to write and calculate large numbers. Instead of writing \(2 \times 2 \times 2\) five times, you can simply write \(2^5\). This saves time and reduces the potential for errors.It's crucial to understand how exponents interact with one another, especially with rules like the power of a power, which simplifies computations significantly.

Simplifying Expressions

Simplifying expressions is all about making them as straightforward as possible without changing their value. When dealing with expressions involving exponents, simplification often involves applying specific rules to reduce them.Here are some tips for simplifying expressions with exponents:

• Use the power of a power rule when you see multiple exponents as in \((a^m)^n\). Multiply the exponents to simplify: \(a^{m \times n}\).
• Ensure every part of the expression is written in its simplest form, meaning you perform all possible multiplications or operations.

Simplification makes calculation easier and helps reveal the underlying structure of the expression. In the exercise, you start with \((2^m)^2\) and simplify it to \(2^{2m}\), making it easier to handle in equations, comparisons, or additional computations. Understanding how to simplify ensures clarity and efficiency when working with exponents.