Problem 46 - Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (\sqrt[3]{9})^{3} $$

Step-by-Step Solution

Answer

The simplified result is 9.

Step 1: Understand the Expression

The expression \((\sqrt[3]{9})^{3}\) asks us to take the cube root of 9 and then raise the result to the power of 3. This is a simplification of the expression.

Step 2: Apply the Cube Root and Power

The cube root \(\sqrt[3]{9}\) is a number that, when cubed, gives 9. When we raise \(\sqrt[3]{9}\) to the power of 3, it effectively undoes the cube root. Mathematically: \((\sqrt[3]{9})^{3} = 9\), because \((x^{1/3})^{3} = x\) for any \(x\).

Step 3: Simplify the Expression

Since \((\sqrt[3]{9})^{3} = 9\), the expression simplifies directly to 9.

Key Concepts

Exponent Rules

Exponent rules are fundamental tools in algebra that help us simplify complex mathematical expressions. An important exponent rule to remember is

• the rule that states \[ (x^{m})^{n} = x^{m \cdot n}. \]

This means that when you raise a power to another power, you multiply the exponents.

In the context of our given expression \[ (\sqrt[3]{9})^{3}, \]we can substitute \(\sqrt[3]{9} \text{ with } 9^{1/3} \). Using our exponent rule, plugging \(9^{1/3}\) into \[ (9^{1/3})^{3} \]results in \[ 9^{(1/3)\cdot 3} = 9^{1}. \]This simplifies directly to 9 according to the rules of exponents, confirming the underlying operation is mathematically sound.

Understanding exponent rules simplifies the effort needed to solve expressions and helps in efficiently solving complex algebraic problems.

Cube Roots

The cube root of a number is one of its three equal factors. Mathematically, if

• \( y = \sqrt[3]{x} \),

then it implies

• \( y^{3} = x \).

Cube roots are essential in simplifying expressions, especially those involving exponents.

For example, in the expression \[ (\sqrt[3]{9})^{3}, \]the cube root of 9 is a value that when cubed, results in 9. By the nature of cube roots, this expression consists of performing two inverse operations: taking a cube root and then cubing it. These process effectively cancel each other out, leading us back to the original number, 9.

Cube roots might look challenging initially, but they are very straightforward once you understand they simply reverse cubing a number.

Simplifying Expressions

Simplifying expressions involves combining and reducing numbers and operations into a clean, straightforward result. This process not only makes calculations easier but also enhances our understanding of the problem at hand.

• The goal is to break down complex expressions into simpler forms, which often involve adhering to mathematical conventions and rules such as those for exponents.
• The simplification might involve expanding, factoring, and combining like terms, among other operations.

In the expression \[ (\sqrt[3]{9})^{3}, \]we simplify by recognizing the inverse relationship between cubing and cube rooting. This realization allows us to reduce the expression directly to 9 without further computation.

Mastering simplification is an invaluable skill, enabling quicker solutions and insightful deductions in more advanced mathematical problems.