Problem 41 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ (\sqrt[3]{5 z}+\sqrt[3]{3})(\sqrt[3]{5 z}+2 \sqrt[3]{3}) $$

Step-by-Step Solution

Answer

\(\sqrt[3]{25z^2} + 3\sqrt[3]{15z} + 2\sqrt[3]{9}\)

Step 1: Apply the Distributive Property

Use the distributive property to expand the expression \((\sqrt[3]{5z} + \sqrt[3]{3})(\sqrt[3]{5z} + 2\sqrt[3]{3})\). This means multiplying each term in the first parenthesis by each term in the second parenthesis:Let \(a = \sqrt[3]{5z}\) and \(b = \sqrt[3]{3}\):\[a(a+2b) = a^2 + 2ab + b(a+2b)\] \[ = a^2 + 2ab + ab + 2b^2 \] Thus, \[ (\sqrt[3]{5z})^2 + 2(\sqrt[3]{5z})(\sqrt[3]{3}) + (\sqrt[3]{3})(\sqrt[3]{5z}) + 2(\sqrt[3]{3})^2\].

Step 2: Simplify Each Term

We simplify each term from the expression:1. \((\sqrt[3]{5z})^2 = \sqrt[3]{(5z)^2} = \sqrt[3]{25z^2}\).2. \(2\sqrt[3]{5z}\sqrt[3]{3} = 2\sqrt[3]{5z \cdot 3} = 2\sqrt[3]{15z}\).3. \(\sqrt[3]{3}\cdot\sqrt[3]{5z} = \sqrt[3]{3 \cdot 5z} = \sqrt[3]{15z}\) (which combines with the term from above).4. \(2(\sqrt[3]{3})^2 = 2\sqrt[3]{3^2} = 2\sqrt[3]{9}\).

Step 3: Combine Like Terms

Combine the like terms from the expression:1. Combine \(2\sqrt[3]{15z}\) and \(\sqrt[3]{15z}\) to get \(3\sqrt[3]{15z}\).The expression simplifies to: \[\sqrt[3]{25z^2} + 3\sqrt[3]{15z} + 2\sqrt[3]{9}\]

Step 4: Final Simplified Expression

The fully expanded and simplified expression is:\[\sqrt[3]{25z^2} + 3\sqrt[3]{15z} + 2\sqrt[3]{9}\] Each term is expressed in simplest form based on the given variables.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers where we multiply a number (the base) by itself a certain number of times, as indicated by the exponent. The expression can be expressed as \( a^n \), where \( a \) is the base and \( n \) is the exponent. The process greatly helps in simplifying expressions and equations.
For instance:

• \( 2^3 = 2 \times 2 \times 2 = 8 \)
• \( 5^2 = 5 \times 5 = 25 \)

In the context of our exercise, exponentiation appears when we square the cube roots, such as in \((\sqrt[3]{5z})^2\). Simplifying the exponents and roots is an integral step in manipulating algebraic expressions.

Cube Roots

A cube root is a number that, when multiplied by itself twice, equals the given number. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). This concept extends naturally to algebraic terms, where it becomes a useful tool for breaking down polynomial expressions.
In mathematical notation, the cube root of a number \( x \) is expressed as \( \sqrt[3]{x} \). Understanding cube roots is crucial in this exercise because they are used extensively within both the terms of the expression and their simplifications:

• \((\sqrt[3]{5z})^2\)
• \(2\sqrt[3]{5z} \cdot \sqrt[3]{3}\)
• \(2(\sqrt[3]{3})^2\)

Knowing how to manipulate these roots helps you simplify the expression accurately.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition, subtraction, multiplication, and division. They form the building blocks of algebra and are key to understanding how to manipulate equations and functions.
In the exercise, our expression is \((\sqrt[3]{5z} + \sqrt[3]{3})(\sqrt[3]{5z} + 2\sqrt[3]{3})\). Each part of this expression can be analyzed individually and then combined using operations to simplify the entire expression. Using variables helps in generalizing the mathematical concepts, allowing for a broader application.Whenever you encounter algebraic expressions:

• Identify and group like terms.
• Use the distributive property where necessary to expand or simplify expressions.
• Ensure any operations follow the correct order: parentheses, exponents, multiplication/division, addition/subtraction.

These principles enable you to tackle complex algebraic problems efficiently.

Polynomial Simplification

Polynomial simplification involves breaking down complex polynomial equations into simpler forms. This is crucial for revealing solutions to equations or gaining insights about their behavior.
The simplification process usually involves combining like terms and applying algebraic rules like the distributive property. In the provided exercise, simplifying the polynomial expression is a multi-step process:

• Use the distributive property to expand the expression \((\sqrt[3]{5z} + \sqrt[3]{3})(\sqrt[3]{5z} + 2\sqrt[3]{3})\).
• Combine like terms, ensuring each step maintains the expression's integrity.
• Simplify each term individually using exponentiation and cube root rules, before combining them.

By following these methods, you achieve a simplified form, which often reveals the nature of the expression in a more digestible format. Understanding polynomial simplification is vital for tackling more complex algebraic problems.