Problem 72 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\(\sqrt[3]{6x^2} - 3x\sqrt[3]{3x} - \sqrt[3]{3x} + 3\sqrt[3]{2x} - 9x - 3\) is the simplified expression.

Step 1: Expand the Expression

The given exercise requires us to multiply the binomials \((\sqrt[3]{3x} + 3)\) and \((\sqrt[3]{2x} - 3x - 1)\). Start by using the distributive property to expand the expression: \[(\sqrt[3]{3x})(\sqrt[3]{2x}) + (\sqrt[3]{3x})(-3x) + (\sqrt[3]{3x})(-1) + 3(\sqrt[3]{2x}) + 3(-3x) + 3(-1)\]

Step 2: Simplify Each Product

Calculate each of the products from the expansion to simplify the expression:1. The product \((\sqrt[3]{3x})(\sqrt[3]{2x})\) becomes \(\sqrt[3]{6x^2}\).2. The product \((\sqrt[3]{3x})(-3x)\) becomes \(-3x\sqrt[3]{3x}\).3. The product \((\sqrt[3]{3x})(-1)\) becomes \(-\sqrt[3]{3x}\).4. The product \(3(\sqrt[3]{2x})\) becomes \(3\sqrt[3]{2x}\).5. The product \(3(-3x)\) becomes \(-9x\).6. The product \(3(-1)\) is \(-3\).

Step 3: Combine Like Terms

After calculating each product, combine like terms if possible:\[\sqrt[3]{6x^2} - 3x\sqrt[3]{3x} - \sqrt[3]{3x} + 3\sqrt[3]{2x} - 9x - 3\]There are no like terms in terms of radicals that can be combined further, so this is the simplified form of the expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps in multiplying expressions. This property allows you to multiply a single term by each term within a parenthesis. In mathematics, we often write it as follows:

• For any three numbers, a, b, and c: \( a(b+c) = ab + ac \)

This property is particularly useful in polynomial multiplication as seen in the provided exercise.In the problem, we apply the distributive property to multiply two expressions. Firstly, we take \( \sqrt[3]{3x} + 3 \) and multiply it by each term in the other expression, \( \sqrt[3]{2x} - 3x - 1 \). Let's break this down:

• Multiply \( \sqrt[3]{3x} \) by each term: \( \sqrt[3]{3x}(\sqrt[3]{2x}), \sqrt[3]{3x}(-3x), \text{and}\ \sqrt[3]{3x}(-1) \)
• Then, multiply 3 by each of those terms: \( 3(\sqrt[3]{2x}),\ 3(-3x), \text{and}\ 3(-1) \)

This expansion allows us to calculate each of these smaller products one by one. It provides a systematic approach to ensure that no terms are left out when multiplying two polynomials.

Cube Roots

Cube roots are an important element in various algebraic expressions. They allow us to find a number which, when multiplied by itself twice (or cubed), gives the original number. For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\). In mathematical terms, the cube root of a number a is denoted as \( \sqrt[3]{a} \).
In the context of our exercise, the cube roots play a crucial role during multiplication.When multiplying expressions like \( \sqrt[3]{3x} \) and \( \sqrt[3]{2x} \), there is an interesting interplay. We calculate their product as \( \sqrt[3]{6x^2} \). This happens because you multiply the radicands (the numbers inside the root) together. Hence, \( 3x \times 2x = 6x^2 \), and then take the cube root of that product.Cube roots can be tricky, but with practice, understanding how to manipulate them becomes manageable.Keep in mind:

• They are the inverse operation of cubing a number.
• Multiplying cube roots involves multiplying under the same root.

Exploring cube roots in this manner can deepen your understanding and enhance your confidence in algebraic calculations.

Combining Like Terms

Combining like terms is a key step in simplifying algebraic expressions. "Like terms" refer to terms that have the same variables raised to the same power. For instance, \(3x\) and \(-9x\) are like terms because they both have the variable \(x\) raised to the power of 1.When simplifying the expression from our exercise, combining like terms helps reduce the expression to its simplest form. However, it's critical to identify which terms can actually be combined:
In the provided problem:\[\sqrt[3]{6x^2} - 3x\sqrt[3]{3x} - \sqrt[3]{3x} + 3\sqrt[3]{2x} - 9x - 3\]Each term is examined:

• \( \sqrt[3]{6x^2} \), \( -3x\sqrt[3]{3x} \), \( -\sqrt[3]{3x} \), and \( 3\sqrt[3]{2x} \) are distinct terms due to different radicals or variables.
• \(-9x\) and \(-3\) cannot be combined with any radical terms.

Thus, there are no like terms among these that can be further combined, and the expression remains in its current simplified form.Understanding how to determine and combine like terms can significantly aid in mathematical problem solving and expression simplification.