Problem 69 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{4}+2)(\sqrt[3]{2}-1) $$

Step-by-Step Solution

Answer

\(2\sqrt[3]{2} - \sqrt[3]{4}\)

Step 1: Distribute (FOIL)

\((\sqrt[3]{4}+2)(\sqrt[3]{2}-1) = \sqrt[3]{4} \cdot \sqrt[3]{2} - \sqrt[3]{4} \cdot 1 + 2 \cdot \sqrt[3]{2} - 2 \cdot 1\)

Step 2: Simplify each term

\(\sqrt[3]{4} \cdot \sqrt[3]{2} = \sqrt[3]{8} = 2\)

So the expression becomes: \(2 - \sqrt[3]{4} + 2\sqrt[3]{2} - 2 = 2\sqrt[3]{2} - \sqrt[3]{4}\)

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra, making it easier to multiply expressions. It involves distributing one expression across another by multiplying every term within the first expression with every term in the second one. This property is handy when dealing with polynomials and radicals alike.
This property states that for any three numbers or expressions, say a, b, and c, the equation will always hold:

• \(a(b + c) = ab + ac\)
• \((a - b)c = ac - bc\)

In the case of our problem, the expression is \((\sqrt[3]{4} + 2)(\sqrt[3]{2} - 1)\). The distributive property comes into play as each term from the first expression is multiplied with each from the second:

• \(\sqrt[3]{4} \cdot \sqrt[3]{2}\)
• \(-\sqrt[3]{4} \cdot 1\)
• \(+ 2 \cdot \sqrt[3]{2}\)
• \(- 2 \cdot 1\)

This results in the new expression \(\sqrt[3]{4} \cdot \sqrt[3]{2} - \sqrt[3]{4} + 2 \cdot \sqrt[3]{2} - 2\). Understanding this step is crucial as it lays the foundation for further simplification.

Simplifying Expressions

Once you've applied the distributive property, the next step is to simplify the resulting expression. Simplifying helps in making the expression as concise and manageable as possible.
Let's consider the expression obtained from our problem: \(\sqrt[3]{4} \cdot \sqrt[3]{2} - \sqrt[3]{4} + 2 \cdot \sqrt[3]{2} - 2\).
The key to simplification is to group like terms and reduce the expression by carrying out any possible operations. Here are a few tips:

• Combine like terms: Look for terms that have the same radical base or numerical value. In some cases, it might not be fully possible to simplify further, but every bit helps.
• Calculate products: Multiply radicals when their bases are the same. For example, \(\sqrt[3]{4} \cdot \sqrt[3]{2}\).
• Simplify constants: Evaluate any arithmetic involving constants, like \(-2\).

Simplifying expressions gives clarity to complex problems and helps in further calculations.

Cubic Roots

Cubic roots are the inverse operation of a cube, similar to how square roots work with squares. They are particularly important when simplifying expressions with cubes.A cubic root of a number `a`, denoted \(\sqrt[3]{a}\) or \(a^{1/3}\), is the number `b` such that \(b^3 = a\). For instance:

• The cubic root of 8 is 2 because \(2^3=8\).
• For \(\sqrt[3]{27}\), the answer is 3, since \(3^3=27\).

In expressions involving cubic roots, multiplication can sometimes simplify the terms. For example, in our exercise, \(\sqrt[3]{4} \cdot \sqrt[3]{2}\) can be simplified by multiplying the numbers first: \(\sqrt[3]{8}\). Since \(2^3 = 8\), \(\sqrt[3]{8} = 2\).
Mastering cubic roots is key to handling more complex algebraic expressions and simplifying them to their core values effectively.