Problem 92 - Elementary and Intermediate Algebra

Question

Perform the multiplications. All variables represent positive real numbers. See Example 9. $$ x^{4 / 3}\left(x^{2 / 3}+3 x^{5 / 3}-4\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( 3x^3 + x^2 - 4x^{4/3} \).

Step 1: Distribute the First Term

Our goal is to multiply each term inside the parentheses by the term outside of it. Begin by distributing the term \( x^{4/3} \) to \( x^{2/3} \). Recall that when multiplying terms with the same base, you add the exponents. Thus, \( x^{4/3} \times x^{2/3} = x^{(4/3) + (2/3)} = x^{6/3} = x^2 \).

Step 2: Distribute to the Second Term

Now distribute \( x^{4/3} \) to the second term \( 3x^{5/3} \). Again, add the exponents: \( x^{4/3} \times 3x^{5/3} = 3x^{(4/3)+(5/3)} = 3x^{9/3} = 3x^{3} \).

Step 3: Distribute to the Third Term

Finally, distribute \( x^{4/3} \) to \(-4\). You simply multiply the coefficient since \( x^{-4} \) times \(-4\) leads to \(-4x^{4/3} \).

Step 4: Write the Expression

Combine all distributed terms into a single expression. The expression after multiplication becomes: \( x^2 + 3x^3 - 4x^{4/3} \).

Step 5: Simplify the Expression (if needed)

Re-evaluate each term to ensure no further simplification is needed. However, in this expression, all terms have distinct exponents, so this is the simplest form. Write the final answer: \( 3x^3 + x^2 - 4x^{4/3} \).

Key Concepts

Exponent Rules

Exponent rules form the foundational stone for understanding algebraic expressions. When dealing with exponents, especially in multiplication, it's essential to know how they operate.

• Multiplying with Same Base: If the bases are the same (like both being "x"), you simply add the exponents. For example, when multiplying \( x^{4/3} \) with \( x^{2/3} \), you add the exponents \((4/3 + 2/3)\), resulting in \( x^{6/3} \) or \( x^{2} \).
• Raising a Power to a Power: Although not directly applicable in the given problem, it's vital to remember when you have a power raised to another power, you multiply the exponents.

These rules are practical tools that help simplify complex calculations. Understanding them allows for the easy manipulation of expressions without confusion. By using these rules, you can multiply and simplify expressions with confidence. And remember, these simplification techniques make algebraic solutions not only accurate but also quicker.

Polynomial Multiplication

Polynomial multiplication might seem daunting initially, but it's straightforward once you grasp the basics. It's about distributing terms effectively, ensuring each term is multiplied by others within the expression.First, consider the polynomial structure. A polynomial is an expression made up of variables (like \(x\)) and coefficients, combined using addition or subtraction. When we multiply polynomials:

• You multiply each term in one polynomial by every term in the other.
• Add the results together, keeping note of like terms and simplifying where possible.

For the exercise, we distribute the term \(x^{4/3}\) across each term within the parentheses \((x^{2/3} + 3x^{5/3} - 4)\). This involves applying the distributive property and focusing on adding the exponents for like bases.Using this approach ensures no terms are missed, leading to a correctly expanded expression. This keeps calculations organized, maintaining clarity and accuracy throughout the process.

Simplifying Expressions

Simplifying expressions is crucial for reaching the most concise form of an algebraic expression. This involves combining like terms and putting expressions in a more manageable form.Here's how to simplify effectively:

• Identify Like Terms: Look for terms with the same base and exponent. These can often be simplified or combined.
• Perform All Possible Arithmetic: Look through the expression for any arithmetic simplifications or factorizations you can do. But be cautious and ensure every possible simplification rule is applied appropriately.
• Maintain Accuracy: Constantly double-check each step to avoid errors. This means re-evaluating each term to confirm no further simplification is possible, as seen in the final step of the exercise where the answer becomes \( 3x^3 + x^2 - 4x^{4/3} \).

Ultimately, simplifying expressions makes them easier to interpret and use, whether for further algebraic operations or understanding their behavior. It turns complex equations into easily solved or understood components, facilitating better mathematical intuition and problem-solving skills.