Problem 73 - A Graphical Approach to College Algebra

Question

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$(\sqrt[3]{11}-1)(\sqrt[3]{11^{2}}+\sqrt[3]{11}+1)$$

Step-by-Step Solution

Answer

The simplified expression is 10.

Step 1: Recognize the Expression Pattern

The expression \((\sqrt[3]{11}-1)(\sqrt[3]{11^{2}}+\sqrt[3]{11}+1)\) fits the pattern \((a-b)(a^2+ab+b^2)\) which is equivalent to \(a^3-b^3\). Here, \(a=\sqrt[3]{11}\) and \(b=1\). This recognition helps in simplifying the expression.

Step 2: Calculate the Cubes

Since \(a = \sqrt[3]{11}\) and \(b = 1\), calculate \(a^3\) and \(b^3\): - \(a^3 = (\sqrt[3]{11})^3 = 11\) - \(b^3 = 1^3 = 1\).

Step 3: Simplify the Expression

Apply the identity \((a-b)(a^2+ab+b^2) = a^3 - b^3\). Thus, the expression simplifies to \(a^3 - b^3 = 11 - 1 = 10\).

Key Concepts

Real Numbers

Real numbers are the set of numbers that include both rational and irrational numbers. They can be represented on a number line and play a crucial role in algebra, calculus, and many other areas of mathematics. Essentially, real numbers include:

• Integers: Whole numbers and their negatives, like \(-3, 0, 2\).
• Rational numbers: Numbers that can be expressed as the quotient of two integers, like \(\frac{2}{3}\) or \(-5\).
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, such as \(\sqrt{2}\) or \(\pi\).

Every real number has a corresponding point on the number line. This trait makes them essential for understanding and applying concepts across mathematics. Understanding that variables in expressions represent real numbers allows us to perform operations and simplifications knowing the numbers are within this comprehensive set.

Cubic Roots

The cubic root of a number is a value that, when multiplied by itself twice (three times in total), gives the original number. This can be represented as \(\sqrt[3]{x}\). For example, \(\sqrt[3]{8} = 2\) because \(2 \times 2 \times 2 = 8\). Recognizing and calculating cubic roots are important skills in solving and simplifying algebraic expressions.

In the context of our exercise, we dealt with the cubic root of 11, represented as \(\sqrt[3]{11}\). Calculating the cube of this expression involves understanding that \(\sqrt[3]{11}^3 = 11\), since we are essentially reversing the root process.

Understanding cubic roots and their properties is essential when dealing with higher degree polynomials and solving complex algebraic equations.

Expression Patterns

Identifying expression patterns allows us to simplify complex problems with ease. The given exercise fits the pattern \((a-b)(a^2+ab+b^2) = a^3-b^3\). This pattern is a specific case of factoring called a sum or difference of cubes. It simplifies the operations by recognizing the structure and knowing an identity that relates these algebraic forms.

• Pattern Recognition: By identifying \(a = \sqrt[3]{11}\) and \(b = 1\), we utilize this pattern directly.
• Application of Identity: Simplifying \((a-b)(a^2+ab+b^2)\) helps us quickly find that it equals \(a^3-b^3\).

Such pattern recognition streamlines the process by avoiding lengthy multiplication and instead using algebraic identities to arrive at solutions like \(11 - 1 = 10\). Recognizing and applying these patterns are fundamental skills in algebra, helping to simplify expressions accurately and efficiently.