Problem 11 - Algebra 1: Concepts and Skills

Question

Use the power of a product property to simplify the expression. \((2 n)^{4}\)

Step-by-Step Solution

Answer

The simplification of the algebraic expression \((2n)^4\) using the power of a product property is \(16n^4\).

Step 1: Apply Power of a Product Property

Here, one needs to apply the power of a product property to the expression \((2n)^4\). According to the power of a product property, (ab)^n = a^n * b^n. So, \((2n)^4\) can be rewritten as \(2^4 * n^4\).

Step 2: Simplify Further

Next, simplify the expression further by calculating the power of 2. In other words, work out what \(2^4\) equals. This calculation leaves the algebraic description as \(16 * n^4\).

Step 3:Final Simplification

Finally the simplified version of the original expression \((2n)^4\) is written as \(16n^4\). This step doesn't involve much more than stating the already known result, therefore making it highly straightforward.

Key Concepts

Simplifying Algebraic Expressions

In algebra, simplifying expressions is the process of making them easier to understand or work with. This can involve a variety of techniques, including combining like terms, using the distributive property, or applying exponent rules as we see with the power of a product property.

Often, an algebraic expression can seem daunting, but by breaking it down step by step—like turning \( (2n)^4 \) into \( 16n^4 \)—we make the problem more manageable. The goal is always to write the expression in its simplest form without changing its value. For students, mastering how to simplify expressions is crucial, as it's a foundational skill used in solving equations and manipulating algebraic formulas.

Exponents and Powers

Exponents are a shorthand way of expressing repeated multiplication of the same factor. In the expression \(2^4\), the number 2 is the base, and the exponent is 4, signifying that 2 is multiplied by itself 4 times. Powers, like \(2^4\), simplify to \(2 \times 2 \times 2 \times 2\) or 16.

Understanding how to work with exponents is important for multiple areas in mathematics, including algebra, where they frequently appear. The power of a product, like \( (2n)^4 \), shows how exponentiation applies not just to a single number, but also to an expression consisting of a product of factors. Grappling with this concept means learning to apply the exponent to each factor within the product, as shown in the solution to the provided exercise.

Algebraic Properties

Algebraic properties are rules that describe how numbers and expressions can be manipulated without changing their underlying relationships or values. One such property is the power of a product property, which states that the power of a product \( (ab)^n \) is equal to the product of the powers \( a^n \times b^n \).

In practice, this principle allows students to take an expression with a power applied to a product, like \( (2n)^4 \), and simplify it by separately raising each factor in the product to the power, resulting in \( 2^4 \times n^4 \). Each algebraic property, like the power of a product, is a tool meant to aid in the simplification process, transforming complex expressions into simpler ones to work with. Understanding and applying these properties are essential skills in algebra.