Problem 44 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[3]{27+a^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \( 3 + a \).

Step 1: Recognize the Form

The expression we have is \( \sqrt[3]{27 + a^3} \). Notice that this expression resembles a sum of cubes: \( 27 \) is \( 3^3 \) and \( a^3 \) is already a cube. To simplify, we can use the identity for the sum of cubes: \( x^3 + y^3 = (x+y)(x^2 - xy + y^2) \).

Step 2: Identify Components

Here, \( x = 3 \) and \( y = a \). Then the expression \( 27 + a^3 \) can be rewritten using the identity: \( (3 + a)(9 - 3a + a^2) \).

Step 3: Apply the Cube Root

Apply the cube root to the expression: \( \sqrt[3]{(3 + a)(9 - 3a + a^2)} \). Since \( \sqrt[3]{x^3} = x \) for any positive real \( x \), \( \sqrt[3]{27 + a^3} = 3 + a \).

Step 4: Final Simplified Expression

Combine everything to conclude that the simplified expression for \( \sqrt[3]{27 + a^3} \) is \( 3 + a \).

Key Concepts

Simplifying Expressions

Simplifying radical expressions might seem tricky at first, but with practice, you'll get the hang of it. The key is to transform complex expressions into simpler or more workable forms without changing their value. Here's a breakdown of what this means:

• Identify patterns or familiar forms, such as squares or cubes, within the expression.
• Use identities or formulas to rewrite the expression in simpler components.

Consider the exercise: \(\sqrt[3]{27 + a^3}\).
Recognizing that 27 is \(3^3\) and \(a^3\) is already a cube can simplify the process. By relating them to known identities, the expression becomes manageable. This leads to expressing it as a sum of cubes, which we'll explore next. The simplification results in a cleaner and much more intuitive expression: \(3 + a\). This approach not only highlights the power of algebraic identities but also builds a deeper understanding of mathematical structures.

Sum of Cubes

The sum of cubes is a special algebraic identity that can simplify complex expressions. The identity for sum of cubes is:

\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \]
In the expression \(27 + a^3\), \(27\) is equivalent to \(3^3\) and \(a^3\) is already one of the cubes. Recognizing these components allows us to rewrite the expression as:

\[ (3 + a)(9 - 3a + a^2) \]
Each factor is derived directly using the sum of cubes formula. Simplifying in this way is straightforward once you find the cubic terms. It unfolds the expression into products that make the final expression \(3 + a\) more digestible. If you remember this identity, you can simplify many similar problems with ease.

Cube Roots

Understanding cube roots is crucial when working with expressions that involve cubes. A cube root is essentially a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\).

Applying cube roots to expressions involves:

• Recognizing when an expression is a perfect cube.
• Simplifying using the relation \(\sqrt[3]{x^3} = x\).

In the exercise, applying the cube root \(\sqrt[3]{(3 + a)(9 - 3a + a^2)}\) to the simplified form \((3 + a)\) draws from this principle. Since taking the cube root of a cube returns the base, the solution simplifies neatly. Thus, the understanding of cube roots forms a pivotal part of simplifying and solving these expressions.