Problem 42 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ (\sqrt[3]{3 p}-2 \sqrt[3]{2})(\sqrt[3]{3 p}+\sqrt[3]{2}) $$

Step-by-Step Solution

Answer

The simplified expression is \((3p)^{2/3} - 2^{2/3}\).

Step 1: Recognize the Formula

The expression given is of the form \((a - b)(a + b)\), which follows the difference of squares pattern: \((a^2 - b^2)\). Here, \(a = \sqrt[3]{3p}\) and \(b = \sqrt[3]{2}\).

Step 2: Apply the Formula

According to the formula for the difference of squares, \((a - b)(a + b) = a^2 - b^2\). Substitute \(a = \sqrt[3]{3p}\) and \(b = \sqrt[3]{2}\) into the formula: \[ (\sqrt[3]{3p})^2 - (\sqrt[3]{2})^2 \]

Step 3: Simplify the Exponentials

Simplify each cubic square: \[ (\sqrt[3]{3p})^2 = (3p)^{2/3} \] and \[ (\sqrt[3]{2})^2 = 2^{2/3} \] Combine these results to express the simplified form: \[ (3p)^{2/3} - 2^{2/3} \]

Step 4: Present the Simplified Expression

The simplified expression is: \[ (3p)^{2/3} - 2^{2/3} \] This is the simplest form of the given product of expressions.

Key Concepts

Understanding Difference of Squares

The difference of squares is a foundational concept in algebra. It’s a special factoring technique used to simplify expressions that follow the pattern \((a - b)(a + b)\). This formula simplifies to \(a^2 - b^2\), which is much easier to compute, especially for large numbers or complex expressions. This simplification process works because:

• The 'minus' and 'plus' in the brackets cause the middle terms to cancel each other out when expanded.
• What remains is only the square of each term within the parentheses.

In our example, the expression \((\sqrt[3]{3p} - \sqrt[3]{2})(\sqrt[3]{3p} + \sqrt[3]{2})\) fits the difference of squares pattern, allowing us to re-write it as \((\sqrt[3]{3p})^2 - (\sqrt[3]{2})^2\). Not only does this simplify calculations but also allows deeper understanding of polynomial identities.

What are Cubic Roots?

Cubic roots refer to finding a number that, when multiplied by itself three times, gives the original number. In mathematical terms, the cubic root of a number \(x\) is denoted as \(\sqrt[3]{x}\). For example, \(\sqrt[3]{27} = 3\), as multiplying 3 three times gives 27.

• Cubic roots are important in solving equations where the variable is raised to the power of three.
• They simplify the expression by making it possible to factorize or rewrite in simpler forms.

In the given exercise, the terms \(\sqrt[3]{3p}\) and \(\sqrt[3]{2}\) are cubic roots, making it essential to understand how they operate when squared or combined in expressions like the difference of squares.

Techniques of Simplification

Simplification involves reducing expressions to their simplest form, making them easier to interpret and solve. This often involves factoring, combining like terms, or utilizing algebraic identities. The step-by-step solution helps in breaking down complex expressions using:

• Formulas such as the difference of squares for factoring.
• Properties of exponents, such as \(a^{m/n} = (a^m)^{1/n}\), to convert roots into fractional exponents.

By applying these strategies, we tackle components of an expression one at a time. In our exercise, \((\sqrt[3]{3p})^2\) simplifies to \((3p)^{2/3}\) because raising a root to a power translates it to a fractional exponent. Simplification hence plays a pivotal role in both understanding and solving algebraic problems efficiently.

Exponential Forms and Operations

Exponents are a compact way to express repeated multiplication of a number by itself. In algebra, understanding how to manipulate them is critical. Exponential notation \(a^b\) indicates that the base \(a\) is multiplied by itself \(b\) times. Important rules include:

• Product of powers: \(a^m \times a^n = a^{m+n}\).
• Power of a power: \((a^m)^n = a^{mn}\).
• Negative exponents: \(a^{-n} = 1/a^n\).

In our context, converting the cubic root expressions \((\sqrt[3]{})\) to their exponential forms allows for easier simplification. Understanding exponent rules enables converting \((\sqrt[3]{3p})^2\) to \((3p)^{2/3}\), aligning computations to standard algebraic practices for further problem-solving.