Problem 72 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[4]{5}+2 \sqrt[4]{5} $$

Step-by-Step Solution

Answer

The simplified expression is \(3 \sqrt[4]{5}\).

Step 1: Identify the Terms

Examine the expression and identify the terms involved. Here, we have two terms: \(\sqrt[4]{5}\) and \(2\sqrt[4]{5}\). Both terms contain \(\sqrt[4]{5}\) as a common factor.

Step 2: Factor Out the Common Term

Since both terms have \(\sqrt[4]{5}\) in common, factor \(\sqrt[4]{5}\) out of the expression: \(1 \times \sqrt[4]{5} + 2 \times \sqrt[4]{5}\).

Step 3: Combine the Coefficients

Add the coefficients of \(\sqrt[4]{5}\), which are 1 and 2. This gives \( (1 + 2) \times \sqrt[4]{5} \).

Step 4: Simplify the Expression

Simplify \((1 + 2)\) to get \(3\), so the expression becomes \(3 \sqrt[4]{5}\).

Key Concepts

Fourth Root

In algebra, roots are a way of reversing exponentiation. The fourth root of a number is what you multiply by itself four times to get that number. For instance, the fourth root of a number \( x \) is denoted as \( \sqrt[4]{x} \). It answers the question: "What number multiplied by itself four times gives \( x \)?"

• Example: The fourth root of 16 is 2, because \( 2 \times 2 \times 2 \times 2 = 16 \).
• Notation: \( \sqrt[4]{x} \) is used to indicate the fourth root.

Recognizing the root function is crucial in many algebraic simplifications and manipulations.

Factoring

Factoring is a powerful algebraic tool that involves breaking down expressions into simpler components, often by finding a common factor. In the context of our expression \( \sqrt[4]{5} + 2\sqrt[4]{5} \), both terms share the common factor \( \sqrt[4]{5} \).

• Identify the common component: Here, it's \( \sqrt[4]{5} \).
• Reform the expression: Rewrite as \( 1 \times \sqrt[4]{5} + 2 \times \sqrt[4]{5} \).
• Use the distributive property: Factor out \( \sqrt[4]{5} \) to get \( (1 + 2) \times \sqrt[4]{5} \).

Factoring simplifies expressions and makes algebraic operations more manageable. Recognizing common factors is a key skill in algebra.

Coefficients

Coefficients are numerical or constant factors in terms of an algebraic expression. In our exercise, they are the numbers multiplied by \( \sqrt[4]{5} \). Understanding coefficients allows you to simplify expressions by combining similar terms.

• Example: In \( 2\sqrt[4]{5} \), the coefficient is 2.
• Combine coefficients when terms share the same base: Add the coefficients 1 and 2 together.
• The resulting expression is \( 3 \times \sqrt[4]{5} \).

Simplifying coefficients by adding, subtracting, or multiplying is a routine process in solving algebraic problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Simplifying these expressions involves recognizing patterns and using various algebraic techniques. For \( \sqrt[4]{5} + 2\sqrt[4]{5} \), we applied several principles to simplify it:

• Identifying and factoring the common component \( \sqrt[4]{5} \).
• Combining like terms using coefficients.
• Simplifying the expression to \( 3\sqrt[4]{5} \).

Working with algebraic expressions requires knowledge of roots, factors, and operations among others, to transform them into more useful or simpler forms. This empowers you to solve equations and understand algebra deeper.