Problem 7 - A Graphical Approach to College Algebra

Question

Simplify each expression. Leave answers with exponents. $$\left(2^{2}\right)^{5}$$

Step-by-Step Solution

Answer

The simplified expression is \(2^{10}\).

Step 1: Identify the power of a power rule

Recognize that the expression \((a^m)^n\) can be simplified using the power of a power rule which states \((a^m)^n = a^{m imes n}\). In this problem, \(a = 2\), \(m = 2\), and \(n = 5\).

Step 2: Apply the power of a power rule

Use the power of a power rule by multiplying the exponents. Therefore, simplify \((2^2)^5 = 2^{2 imes 5}\).

Step 3: Calculate the product of the exponents

Multiply the exponents: \(2 \times 5 = 10\).

Step 4: Write the simplified expression

Use the result from Step 3 to express the simplified form as \(2^{10}\).

Key Concepts

Power of a Power Rule

When dealing with exponents, the **Power of a Power Rule** is a powerful tool that allows us to simplify concepts more easily. This rule addresses situations where you have an exponent raised to another exponent, like \[ (a^m)^n \]. Here's how it works: instead of calculating each exponent step-by-step, you can directly multiply the exponents together. So, \[(a^m)^n = a^{m\times n}\].
In our example, we have \[(2^2)^5\]. Following the Power of a Power Rule, this becomes:

• Identify individual exponents: here, \(m = 2\) and \(n = 5\).
• Multiply them: \(2 \times 5 = 10\).

So, \[(2^2)^5 = 2^{10}\].
By applying this rule, we streamline the process, reducing potential errors from multiple calculations.

Exponent Multiplication

Applying the **Exponent Multiplication** concept is straightforward when the base number remains the same across terms. This is crucial when combining powers of the same base. The rule can be applied as follows:

• If you have the expression \(a^m \times a^n\), where both terms share the same base \(a\), you can simplify it to \(a^{m+n}\).

This rule emerges from the basic definition of exponents as repeated multiplication. By using this principle, the properties of exponents allow you to combine terms efficiently, simplifying otherwise complex expressions.
While this concept wasn't directly applied to our starting problem, understanding it prepares us to manage expressions with multiple bases and exponents across any algebraic operation.

Simplifying Expressions

Simplifying expressions in mathematics often involves using rules that reduce an expression to its simplest or most compact form. In the case of our exercise, this involved: 1. **Recognizing Patterns:** - Spotting that \[(2^2)^5\] fit the Power of a Power framework. 2. **Applying Rules Correctly:** - Using the Power of a Power Rule to multiply exponents correctly. 3. **Performing Arithmetic:** - Calculating the product of the exponents, which is \(2 \times 5 = 10\),leading to \(2^{10}\).
This multi-step process is illustrative of general strategies in simplifying expressions. Breaking down problems into understandable parts with specific rules such as these makes algebra not only manageable but also logical.