Problem 45 - Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (\sqrt[3]{12})^{3} $$

Step-by-Step Solution

Answer

The answer is 12.

Step 1: Understand the Expression

The expression is \((\sqrt[3]{12})^3\), which involves finding the cube of the cube root of 12.

Step 2: Apply the Rule of Cube and Cube Root

Recall that \( (\sqrt[3]{x})^3 = x \). This identity holds because the cube and cube root are inverse operations.

Step 3: Simplify Using the Rule

Apply the identity to the expression: \((\sqrt[3]{12})^3 = 12\).

Key Concepts

Inverse Operations

In mathematics, inverse operations are operations that undo each other. Think of them as actions that "cancel out" the effects of each other. A common example is addition and subtraction. If you add 5 to a number and then subtract 5, you are left with the original number, because addition and subtraction are inverse operations. In the context of the geometric operation of taking cube roots and cubing a number, these two processes also cancel each other out.

Cube rooting a number and then cubing it will bring you back to your original number. The equation \( (\sqrt[3]{x})^3 = x \) perfectly exemplifies this principle. In this formula, taking the cube root and then raising the number to the power of three reverts it back to its initial value \( x \). By understanding inverse operations, you simplify complex problems and solve equations more effectively.

Simplification

Simplification refers to the process of making an equation or expression easier to understand. It's about reducing complexity. When simplifying expressions, you use various mathematical rules and identities to transform them into simpler forms without changing their values.

In the given problem

• We have the expression \((\sqrt[3]{12})^3\).
• Our goal is to simplify this expression to its most basic form.
• By applying the rule for inverse operations, we recognize that cube and cube root are inverse of each other.

Thus, when we simplify \((\sqrt[3]{12})^3\), it becomes simply 12. Understanding simplification helps in breaking down complex mathematical operations into understandable pieces.

Exponentiation

Exponentiation is one of the core operations in mathematics. It involves raising a number or expression to a power. This power, or exponent, represents the number of times the base is multiplied by itself. For example, \(3^2\) means 3 is multiplied by itself once, resulting in 9. The number "two" is the exponent, and "three" is the base.

With the exercise at hand, we deal with the concept by understanding that raising a cube root to a power of three is a specific application of exponentiation. The operation of cubing the cube root

• directly applied gives back the original number.
• Cube root cancels out cubing.

This is why

• the expression \((\sqrt[3]{12})^3\) simplifies to 12.
• Exponentiation here showcases its role in reversing cube rooting, by manifestation of its powerful identity.

Grasping the concept of exponentiation aids tremendously in handling powers and roots efficiently.