Problem 74 - A Graphical Approach to College Algebra

Question

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$(\sqrt[3]{7}+3)(\sqrt[3]{7^{2}}-3 \sqrt[3]{7}+9)$$

Step-by-Step Solution

Answer

The expression simplifies to 34.

Step 1: Identify and Apply the Cubic Formula

Here, we have an expression in the form \[(a+b)(a^2-ab+b^2)=a^3+b^3.\]In our problem, \(a=\sqrt[3]{7}\) and \(b=3\). Thus, the expression \[(\sqrt[3]{7}+3)(\sqrt[3]{7^{2}}-3 \sqrt[3]{7}+9)\]can be simplified using this identity.

Step 2: Calculate the Cubes of a and b

Compute the cubes of \(a\) and \(b\):1. \(a^3 = (\sqrt[3]{7})^3 = 7\).2. \(b^3 = 3^3 = 27\) because the cube of 3 is 27.

Step 3: Apply the Formula to Simplify

Using the cubic identity from Step 1 and substituting the values obtained in Step 2, we simplify the expression:\[a^3 + b^3 = 7 + 27 = 34.\]

Step 4: Conclude with the Simplified Result

The simplified result is just \(34\). The entire expression initially given simplifies down to this single number.

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we aim to make them easier to understand or work with. Simplifying involves reducing the complexity without changing the value of the expression. In mathematics, it's like cleaning up a messy room to make sense of everything in its proper place.

The process of simplifying can involve various operations, such as combining like terms, removing parentheses by distributing, and applying mathematical identities or formulas. A particular focus is on ensuring that the expression is represented in its lowest terms or most straightforward form.

In the context of the example exercise, we see this in action by first recognizing the structure of the expression. Once the cubic identity is identified, the formula is applied to condense the expression from multiple terms to a single number. This approach not only reduces the workload but also helps in clearly understanding the essence of the expression.

Cubic Identity

The cubic identity is a algebraic gem used for factoring and simplifying specific kinds of expressions. This particular identity is noted as:

• \[(a+b)(a^2-ab+b^2)=a^3+b^3\]

This equation connects a set of seemingly complex binomials and trinomials to their simplest cubic form.

In our exercise, identifying that the expression matches this cubic identity allows us to transform the given expression into something much more simple and manageable. Specifically, when we recognize that

• \(a = \sqrt[3]{7}\)
• \(b = 3\)

we understand that simply by applying the formula the whole expression simplifies to

• \(a^3 + b^3\)

Remember, finding and effectively using identities can speed up problem-solving considerably. Practice recognizing these patterns in expressions to leverage mathematical shortcuts cleverly.

Radicals

Radicals refer to expressions involving roots, such as square roots, cube roots, and higher-order roots. Understanding how to work with radicals is essential since they often appear in algebra and calculus problems.

Cubic roots are a type of radical we encounter in our exercise. The cube root of a number \(n\) is written as \(\sqrt[3]{n}\), meaning a value that, when cubed, returns \(n\). In this case, calculating

• \((\sqrt[3]{7})^3\)

recovers the number 7 because cubing a cube root cancels out the radical.

Radicals are simplified by removing perfect squares or cubes from under the root, reducing the term to its simplest form. This ensures that the expression remains accurate yet simpler to work with, a crucial skill when dealing with more complex equations or algebraic problems.