Problem 42 - Elementary and 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 Expression Type

Observe that the given expression \((\sqrt[3]{3p}-2\sqrt[3]{2})(\sqrt[3]{3p}+\sqrt[3]{2})\) resembles the difference of squares formula, which is \((a-b)(a+b) = a^2 - b^2\). In this case, \(a = \sqrt[3]{3p}\) and \(b = \sqrt[3]{2}\).

Step 2: Apply the Difference of Squares Formula

Use the identity \((a-b)(a+b) = a^2 - b^2\) to simplify the expression. We substitute \(a = \sqrt[3]{3p}\) and \(b = \sqrt[3]{2}\) into the formula: \[((\sqrt[3]{3p})^2 - (\sqrt[3]{2})^2)\]

Step 3: Calculate Each Term

First, calculate \((\sqrt[3]{3p})^2\). Since \(\sqrt[3]{3p} = (3p)^{1/3}\), then \((\sqrt[3]{3p})^2 = (3p)^{2/3}\). Similarly, compute \((\sqrt[3]{2})^2 = (2)^{2/3}\).

Step 4: Simplify the Expression

Substitute the calculated values into the simplified expression:\[(3p)^{2/3} - 2^{2/3}\].This is the simplest form of the expression.

Key Concepts

Difference of Squares

The concept of the difference of squares is a cornerstone in algebraic operations and can dramatically simplify complex expressions. When we talk about the difference of squares, we refer to the formula:

• \((a-b)(a+b) = a^2 - b^2\)

This formula is special because it immediately reduces the expression to a simpler form. The main idea is that when you multiply a binomial expression with the same numbers but opposite signs, the middle terms cancel each other out when you expand them.
In the given exercise, we identify the expression

• \((\sqrt[3]{3p} - 2\sqrt[3]{2})(\sqrt[3]{3p} + \sqrt[3]{2})\)

as a candidate for this rule. By matching the general form, where

• \(a = \sqrt[3]{3p}\)
• \(b = \sqrt[3]{2}\)

the product simplifies directly to:

• \((\sqrt[3]{3p})^2 - (\sqrt[3]{2})^2\).

This remarkable identity saves time and effort in calculations, simplifying problems that might otherwise require more extensive algebraic manipulation.

Cube Roots

Cube roots are another useful algebraic tool. They reverse the operation of cubing a number, i.e., finding the cube root of a number gives a value which, when raised to the power of three, returns the original number. Mathematically, the cube root of a number \(x\) is denoted as \(\sqrt[3]{x}\) or \(x^{1/3}\).
In expressions involving cube roots, such as

• \(\sqrt[3]{3p}\)
• \(\sqrt[3]{2}\)

we often work with their properties to simplify expressions by computing powers of cube roots:

• \((\sqrt[3]{3p})^2 = (3p)^{2/3}\)
• \((\sqrt[3]{2})^2 = (2^{\phantom{t}})^{2/3}\).

Visualize cube roots as fractional exponents. They allow smooth transitions between radicals and exponents, giving more flexibility in solving or transforming expressions.
In our exercise, cube roots were essential in expressing each factor in a way that conforms with the difference of squares pattern, paving the way for further simplification.

Simplification

Simplification in algebra is the process of rewriting expressions in a simpler and more compact form while retaining their original value. It makes solving equations and performing algebraic manipulations easier.
For the given problem, after identifying the expression type and utilizing the difference of squares technique, the bulky original expression becomes the more manageable

• \((3p)^{2/3} - 2^{2/3}\).

The aim is always to present expressions in their simplest form, which makes further calculations or comparisons more straightforward.
Steps to simplification typically involve:

• Substituting known values or identities
• Applying simplification rules (e.g., difference of squares)
• Converting complex terms to simpler equivalents (like transforming radicals through exponential notation)

Through simplification, complex multi-stage calculations become single-step evaluations, making the entire mathematical exercise much less daunting.