Problem 56 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (\sqrt[3]{a}+2)(\sqrt[3]{a}+7) $$

Step-by-Step Solution

Answer

The simplified expression is \(a^{2/3} + 9\sqrt[3]{a} + 14\).

Step 1: Apply the Distributive Property

First, distribute each term in the first binomial \((\sqrt[3]{a} + 2)\) to each term in the second binomial \((\sqrt[3]{a} + 7)\). This means we multiply \(\sqrt[3]{a}\) by both \(\sqrt[3]{a}\) and 7, and then multiply 2 by both \(\sqrt[3]{a}\) and 7.

Step 2: Multiply \(\sqrt[3]{a}\) by each term in the second binomial

Calculate \(\sqrt[3]{a} \times \sqrt[3]{a}\) which equals \((\sqrt[3]{a})^2\) or \(a^{2/3}\). Next, multiply \(\sqrt[3]{a} \times 7\) which equals \(7\sqrt[3]{a}\).

Step 3: Multiply 2 by each term in the second binomial

Calculate \(2 \times \sqrt[3]{a}\) which equals \(2\sqrt[3]{a}\). Then multiply \(2 \times 7\) which equals 14.

Step 4: Combine the terms

Combine all the expressions from Steps 2 and 3: \((a^{2/3} + 7\sqrt[3]{a} + 2\sqrt[3]{a} + 14)\).

Step 5: Simplify the expression

Combine like terms: Add the \(\sqrt[3]{a}\) terms. \(7\sqrt[3]{a} + 2\sqrt[3]{a} = 9\sqrt[3]{a}\). Thus, the simplified expression is \(a^{2/3} + 9\sqrt[3]{a} + 14\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra to simplify multiplication expressions. It allows us to multiply a single term by two or more terms inside parentheses. The basic idea is that you "distribute" the multiplying term to each term inside the parentheses, one at a time. For example, if we have \(x(m + n)\), we distribute \(x\) by multiplying it with both \(m\) and \(n\), resulting in \(xm + xn\).
This property is particularly useful when dealing with polynomial multiplication, like in our exercise. Here, each term of the first binomial \(\left(\sqrt[3]{a} + 2\right)\) is multiplied with each term of the second binomial \(\left(\sqrt[3]{a} + 7\right)\).

• First, multiply \(\sqrt[3]{a}\) by both \(\sqrt[3]{a}\) and \(7\).
• Next, multiply \(2\) by both \(\sqrt[3]{a}\) and \(7\).

This step ensures all combinations of terms are accounted for, paving the way for the simplification process.

Simplification

Simplification is the process of combining and reducing expressions to make them as straightforward as possible. After applying the distributive property, you'll often end up with several terms that can be simplified. In our exercise, once we distribute the terms, we have the expression: \(a^{2/3} + 7\sqrt[3]{a} + 2\sqrt[3]{a} + 14\).
To simplify, we look for like terms. Like terms are those terms that contain the same variable raised to the same power.

• In our expression, \(7\sqrt[3]{a}\) and \(2\sqrt[3]{a}\) are like terms because they both contain \(\sqrt[3]{a}\).
• Adding these together, we get \(9\sqrt[3]{a}\).

Now, our simplified expression becomes \(a^{2/3} + 9\sqrt[3]{a} + 14\). Simplification helps to make expressions cleaner and often reveals the most compact form of a polynomial.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. These can initially seem complex, but they follow specific rules similar to regular numbers. In our exercise, we work with the cube root radical expression \(\sqrt[3]{a}\).
When multiplying radical expressions, there are special considerations:

• For instance, \(\sqrt[3]{a} \times \sqrt[3]{a}\) results in \(a^{2/3}\) because multiplying cube roots results in raising the base to the cumulative power.
• Similarly, \(b^m \times b^n = b^{m+n}\) when working with exponents. The same rule applies to radical multipliers.

It's essential to remember that simplifying radical expressions fully requires not only multiplying and adding them appropriately but also recognizing when they can be further reduced into simpler terms. Understanding radical expressions empowers you to tackle problems involving non-standard powers with confidence.