Problem 55 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (\sqrt[3]{a}-4)(\sqrt[3]{a}+5) $$

Step-by-Step Solution

Answer

\(\sqrt[3]{a^2} + \sqrt[3]{a} - 20\)

Step 1: Expanding the Expression

To expand the expression \((\sqrt[3]{a}-4)(\sqrt[3]{a}+5)\),we apply the distributive property, also known as the FOIL method for two binomials: \(F (First), O (Outer), I (Inner), L (Last)\). This gives us:\(\sqrt[3]{a}\times\sqrt[3]{a} + \sqrt[3]{a}\times 5 -4\times \sqrt[3]{a} -4\times 5\).

Step 2: Calculating Each Term

Calculate each part of the expanded expression separately:1. \(\sqrt[3]{a}\times\sqrt[3]{a} = \sqrt[3]{a^2}\)2. \(\sqrt[3]{a}\times 5 = 5\sqrt[3]{a}\)3. \(-4\times \sqrt[3]{a} = -4\sqrt[3]{a}\)4. \(-4\times 5 = -20 \).

Step 3: Combining Like Terms

Now, combine the like terms from the expanded expression:\(\sqrt[3]{a^2} + 5\sqrt[3]{a} -4\sqrt[3]{a} - 20\). The terms \(5\sqrt[3]{a}\) and \(-4\sqrt[3]{a}\) are like terms:\(\sqrt[3]{a^2} + (5-4)\sqrt[3]{a} - 20\).

Step 4: Simplifying the Expression

Combine the like terms to simplify the expression:\(\sqrt[3]{a^2} + 1\sqrt[3]{a} - 20\).No further simplification is possible since the terms are not combinable.

Key Concepts

Distributive Property

When facing algebraic expressions involving multiplication of binomials, the distributive property becomes a valuable tool. You might recognize it for its role in handling expressions where each term in one binomial is multiplied by each term in the other binomial.
This property simplifies the process of expansion. It follows the format:

• Distribute each term of the first binomial across every term of the other binomial.
• Combine the products obtained from these multiplications.

In our example, distributive property aids in breaking down \((\sqrt[3]{a}-4)(\sqrt[3]{a}+5)\) into smaller terms that are easier to manage and solve. By applying this property, you ensure every interaction between terms is considered, making it foundational in algebraic manipulation.

FOIL Method

The FOIL method is a streamlined version of the distributive property specifically for two binomials. Designed as a simple mnemonic, FOIL stands for First, Outer, Inner, and Last.
Here's how it works:

• **First**: Multiply the first terms of each binomial. In the example, this is \( \sqrt[3]{a} \times \sqrt[3]{a} \).
• **Outer**: Multiply the outer terms. Here, it's \( \sqrt[3]{a} \times 5 \).
• **Inner**: Multiply the inner terms, which are \( -4 \times \sqrt[3]{a} \).
• **Last**: Multiply the last terms, or \( -4 \times 5 \).

Combining these calculations results in the fully expanded form of the expression. It helps to ensure that you cover all possible products of the terms involved, avoiding common algebraic errors.
Use the FOIL method to effortlessly tackle binomial multiplications like a pro, and build a strong foundation in algebraic operations.

Like Terms

After expanding expressions, recognizing like terms becomes crucial in simplifying them. Like terms are terms that have the same variables raised to the same powers. This concept allows us to combine and reduce expressions efficiently.
In the expression we expanded, \(5\sqrt[3]{a}\) and \(-4\sqrt[3]{a}\) are considered like terms. They share the variable component \(\sqrt[3]{a}\).

• Combine them by performing the arithmetic operation on their coefficients (in this case, 5 and -4).
• The result here simplifies to \(1\sqrt[3]{a}\), which we write simply as \(\sqrt[3]{a}\).

Identifying and combining like terms is essential to simplify algebraic expressions further and make them more manageable.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is often the end goal of many math problems. It involves transforming an expression into its most reduced and understandable form.
Simplification is achieved by:

• Expanding expressions using properties like distributive property and FOIL.
• Combining like terms to reduce complexity.
• Ensuring that no further arithmetic or algebraic operations can simplify the expression further.

In the given exercise, after identifying and combining like terms, we arrive at the simplified form \(\sqrt[3]{a^2} + \sqrt[3]{a} - 20\).
Simplicity in algebra helps to clarify the relationships between different parts of an expression, making it easier to understand and apply in real-world problems or more complex mathematical contexts.