Problem 74 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 5 \sqrt[3]{x}-3 \sqrt[3]{x} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2 \sqrt[3]{x} \).

Step 1: Identify Like Terms

In the expression \( 5 \sqrt[3]{x} - 3 \sqrt[3]{x} \), both terms are like terms because they both contain the same cube root \( \sqrt[3]{x} \). They differ only in their coefficients.

Step 2: Combine Coefficients

To simplify the expression, we combine the coefficients of the like terms. Subtract the coefficients: \( 5 - 3 = 2 \). Because the cube root remains the same, the simplified expression becomes \( 2 \sqrt[3]{x} \).

Key Concepts

Cube Roots

Cube roots are one of the basic operations we use when dealing with expressions and equations, especially when simplifying them. The cube root of a number is another number that, when multiplied by itself three times, gives the original number. It is symbolized by the radical sign followed by a small three, as seen in \( \sqrt[3]{x} \).

In the context of algebraic expressions, cube roots can involve variables, such as \( \sqrt[3]{x} \). This indicates that we are looking for a number that, when cubed, results in the value of \( x \).

Understanding cube roots is key in solving problems involving radicals and can also play a significant role in higher-level math topics like polynomial equations.

• The cube root of a positive number results in a positive number.
• Similarly, the cube root of a negative number also results in a negative number, which differs from square roots where negatives are not real numbers.
• Cube roots help simplify expressions like \( 5 \sqrt[3]{x} - 3 \sqrt[3]{x} \) by allowing us to treat \( \sqrt[3]{x} \) as a common factor.

Being comfortable with cube roots will aid significantly in simplifying expressions and solving many algebra-related problems.

Like Terms

In algebra, like terms are terms that have the same variable part, meaning they have exactly the same variables raised to the same power. For example, in the expression \( 5 \sqrt[3]{x} - 3 \sqrt[3]{x} \), both terms are considered like terms, because they contain the same variable and radical part \( \sqrt[3]{x} \).

Identifying like terms is crucial for simplifying expressions. You can combine like terms by adding or subtracting their coefficients, otherwise known as the numeric multiplier of the terms.

• Only the coefficients are combined; the variable part remains unchanged.
• This is because the variable part acts as a common factor, much like variables in simpler algebraic expressions.

Recognizing like terms is a useful skill when simplifying more complex algebraic expressions, allowing for expressions to be condensed and clarified. It makes computations simpler and paves the way for more efficient problem-solving in algebra.

Coefficients

The notion of coefficients is an essential concept in algebra that helps in manipulating and simplifying expressions. A coefficient is the numerical factor that multiplies the variable part of an expression. For example, in the term \( 5 \sqrt[3]{x} \), the number 5 is the coefficient.

Coefficients play a crucial role, especially when working with like terms. When you have like terms, you can simplify the expression by performing operations on the coefficients. This is evident in the expression \( 5 \sqrt[3]{x} - 3 \sqrt[3]{x} \), where the calculation involves subtracting the coefficients 5 and 3.

Key things to remember about coefficients include:

• Coefficients can be positive or negative numbers.
• Operations between like terms revolve solely around the coefficients, while the variable part remains as a common factor.
• Understanding coefficients allows you to simplify expressions effectively and is fundamental in solving equations.

Familiarity with coefficients, like understanding cube roots and like terms, is vital for mastering algebra and can be built upon for advanced mathematical study.