Problem 56 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt[3]{a}+2)(\sqrt[3]{a}+7) $$

Step-by-Step Solution

Answer

The product simplifies to \(\sqrt[3]{a}^2 + 9\sqrt[3]{a} + 14\).

Step 1: Identify the Expression

We need to multiply the two binomials: \((\sqrt[3]{a}+2)(\sqrt[3]{a}+7)\).

Step 2: Apply the Distributive Property

Use the distributive property, also known as the FOIL method for binomials, to multiply \((\sqrt[3]{a}+2)\) and \((\sqrt[3]{a}+7)\), which gives us four terms: \((\sqrt[3]{a})(\sqrt[3]{a}) + (\sqrt[3]{a})(7) + (2)(\sqrt[3]{a}) + (2)(7)\).

Step 3: Simplify Each Term

Calculate each of the four terms from the previous step: 1. \((\sqrt[3]{a})(\sqrt[3]{a}) = \sqrt[3]{a}^2\)2. \((\sqrt[3]{a})(7) = 7\sqrt[3]{a}\)3. \((2)(\sqrt[3]{a}) = 2\sqrt[3]{a}\)4. \((2)(7) = 14\)

Step 4: Combine Like Terms

Combine the like terms in the expanded expression: \(\sqrt[3]{a}^2 + 7\sqrt[3]{a} + 2\sqrt[3]{a} + 14\)Combine the middle terms: \(\sqrt[3]{a}^2 + 9\sqrt[3]{a} + 14\).

Step 5: Final Solution

The expression simplifies to \(\sqrt[3]{a}^2 + 9\sqrt[3]{a} + 14\). This is already simplified as it cannot be factored further using integers.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics that helps us break down complex expressions into simpler parts. It dictates how to multiply a single term by a sum or difference. In the exercise, we are using this property to multiply two binomials: \((\sqrt[3]{a}+2)(\sqrt[3]{a}+7)\).

When using the distributive property with binomials, we often reference the FOIL method, which stands for First, Outer, Inner, and Last. This method helps ensure that each term in the first binomial is multiplied by each term in the second binomial.

• **First**: Multiply the first terms of each binomial. In our case, it's \(\sqrt[3]{a} \times \sqrt[3]{a}\).
• **Outer**: Multiply the outer terms. Here, that's \(\sqrt[3]{a} \times 7\).
• **Inner**: Multiply the inner terms. For us, it's \(2 \times \sqrt[3]{a}\).
• **Last**: Multiply the last terms. This is \(2 \times 7\).

By following this process, you apply the distributive property correctly, ensuring no term is left out of the multiplication, making it a meticulous way to tackle polynomial multiplication.

Binomial Multiplication

Binomial multiplication is a process of multiplying two expressions that each contain two terms. In this exercise, we multiplied \((\sqrt[3]{a}+2)\) with \((\sqrt[3]{a}+7)\).

Each term of the first binomial gets multiplied by each term of the second binomial. Once we apply the distributive property using the FOIL method, we end up with four terms:

• \((\sqrt[3]{a})(\sqrt[3]{a}) = \sqrt[3]{a}^2\)
• \((\sqrt[3]{a})(7) = 7\sqrt[3]{a}\)
• \((2)(\sqrt[3]{a}) = 2\sqrt[3]{a}\)
• \((2)(7) = 14\)

The key to successful binomial multiplication is ensuring each part of one binomial interacts with every part of the other. The order of multiplication follows that of the FOIL approach, but the exact sequence matters less than ensuring consistency and thoroughness in covering all pairwise combinations.

Simplifying Expressions

Simplifying expressions involves combining like terms and making expressions as concise as possible. After multiplying the binomials, we found the expression \(\sqrt[3]{a}^2 + 7\sqrt[3]{a} + 2\sqrt[3]{a} + 14\).

The next step is to look for terms that can be combined:

• **Like Terms:** Terms that share the same variable and same power. In our expression, \(7\sqrt[3]{a}\) and \(2\sqrt[3]{a}\) are like terms.
• **Combine:** Add the coefficients of like terms. So, \(7\sqrt[3]{a} + 2\sqrt[3]{a}\) becomes \(9\sqrt[3]{a}\).

The expression then simplifies to \(\sqrt[3]{a}^2 + 9\sqrt[3]{a} + 14\). Simplification aims to tidy up expressions so they are easier to read and understand, laying a clearer groundwork for any future calculations or expressions.