Problem 19 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(n^{4}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(n^{16}\).

Step 1: Recognize the Power of a Power Rule

When you see an expression of the form \( \left(a^{m}\right)^{n} \), you can apply the power of a power rule. This rule states that \( \left(a^{m}\right)^{n} = a^{m \times n} \).

Step 2: Apply the Rule to the Given Expression

Apply the power of a power rule to the expression \( \left(n^{4}\right)^{4} \). According to the rule: \[ \left(n^{4}\right)^{4} = n^{4 \times 4}. \]

Step 3: Simplify the Exponent Multiplication

Now, multiply the exponents in the expression. Calculate \(4 \times 4\) which equals 16. Thus, the expression becomes \(n^{16}\).

Step 4: Final Expression

The simplified form of the expression is \(n^{16}\). This shows the result of raising \(n\) to the 16th power when simplifying \(\left(n^{4}\right)^{4}\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about making them easier to understand or work with. This involves reducing the complexity of an expression without changing its value.
In the original exercise, the expression \( (n^4)^4 \) was simplified using a clear strategy.

• The Power Rule was recognized and applied correctly.
• Multiplication of like bases was performed to obtain a single term.
• The final result was neatly simplified to \( n^{16} \).

A critical part of simplifying expressions is recognizing patterns or mathematical rules, like the power of a power rule, which can be consistently applied to achieve a simpler form. Simplifying doesn’t alter the overall value—it just represents the same quantity in an easier-to-handle way.

Exponentiation

Exponentiation is a key mathematical operation that describes how many times a number, known as the base, is multiplied by itself. It is denoted by an exponent or power. For example, in \( n^4 \), "n" is the base and 4 is the exponent. Exponentiation allows us to write repeated multiplication in a compact form.
An important concept when dealing with exponentiation is the power of a power rule.
When you have an expression such as \( (a^m)^n \), you don't have to multiply the base itself repeatedly. Instead, multiply the exponents: \( m \) and \( n \). In the given problem, \( (n^4)^4 \) is simplified using this rule: \( n^{4 \times 4} \)which equals\( n^{16} \). This process makes complex expressions more manageable and illustrates how multiplication within exponents simplifies calculations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can consist of numbers, variables, and operators. They are used to represent values in an algebraic form. Each component of the expression plays a vital role. For example:

• Variables, like \( n \), are symbols that represent unknown or changeable quantities.
• Exponents represent repeated multiplication, such as the 4 in \( n^4 \).
• Operators, like multiplication and addition, define the operation being performed between numbers and variables.

In the exercise, \( (n^4)^4 \), iscomposed of a variable \( n \), and the operation rules of exponent addition. Such expressions are pivotal in math as they form the building blocks for solving equations and modeling real-world scenarios. Understanding how to manipulate algebraic expressions is key to simplifying and solving various mathematical problems.