Problem 56 - Elementary and Intermediate Algebra

Question

Use the power rule for exponents to simplify each expression. Write the results using exponents. $$ \left(n^{25}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(n^{100}\).

Step 1: Identify the Power Rule

The power rule for exponents states that when raising a power to another power, you multiply the exponents. This rule can be expressed as \((a^m)^n = a^{m \cdot n}\).

Step 2: Apply the Power Rule

Apply the power rule to the expression \((n^{25})^{4}\). Here, \(a = n\), \(m = 25\), and \(n = 4\). According to the rule, we get \(n^{25 \cdot 4}\).

Step 3: Simplify the Exponents

Calculate the result of multiplying the exponents: \(25 \times 4 = 100\).

Step 4: Write the Final Expression

Replace the expression \((n^{25 \cdot 4})\) with \(n^{100}\). Therefore, \((n^{25})^4 = n^{100}\).

Key Concepts

Exponent Multiplication

Exponent multiplication is a fundamental concept in algebra that involves raising a power to another power. Whenever you have an expression like \((a^m)^n\), you apply the power rule for exponents. This rule states that you multiply the exponents to simplify the expression.

• Identify the base, which is the number or variable being raised to a power. In this case, it is \(n\).
• Look at the exponents involved. For the expression \((n^{25})^4\), you see two exponents: 25 and 4.
• Multiply these exponents: \(25 \times 4 = 100\).

This simplifies the expression to \(n^{100}\). By understanding exponent multiplication, you can quickly simplify complex exponential expressions without performing repeated multiplication.

Simplifying Expressions

Simplifying expressions using the power rule can make seemingly complicated problems much more manageable and provide clearer solutions. Here’s how it works:

• First, identify if your expression uses the power rule by looking for a power raised to another power, like \((n^{25})^4\).
• Apply the rule by multiplying the exponents. In this example, you get \(n^{25 \times 4}\).
• Complete the arithmetic for the exponent multiplication: \(25 \times 4 = 100\).
• The expression becomes \(n^{100}\), which is much simpler and easier to understand.

Simplifying expressions by using this rule can significantly reduce the complexity of the equations you are dealing with, making them easier to work with.

Mathematical Notation

Mathematical notation is like a universal language that enables everyone to understand complex concepts succinctly. When it comes to exponents, several notations are pivotal:

• The base is usually indicated first, such as \(n\) in \(n^{25}\).
• Exponents are written as superscripts, showing the power to which the base is raised.
• For expressions like \((n^{25})^4\), parentheses are used to clarify the order of operations, emphasizing that the base and its exponent are raised to another power.
• Using the power rule, the expression is notated as \(n^{25 \times 4}\) and ultimately simplified to \(n^{100}\).

Mastering the use of mathematical notation allows for efficient communication of mathematical ideas and ensures clarity when performing calculations, such as in exponent multiplication.