Problem 46 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to 9.

Step 1: Understand the Expression

The expression given is \( (\sqrt[3]{9})^{3} \). This means we have the cube root of 9 raised to the power of 3.

Step 2: Apply the Property of Exponents and Roots

Recall that the cube root of a number \( a \), written as \( \sqrt[3]{a} \), is equivalent to \( a^{1/3} \). Therefore, \( (\sqrt[3]{9})^{3} = (9^{1/3})^{3} \).

Step 3: Simplify the Expression Using Exponent Rules

Use the property \( (a^{m})^{n} = a^{m \cdot n} \). Thus, \( (9^{1/3})^{3} = 9^{1/3 \cdot 3} = 9^{1} \).

Step 4: Conclude the Simplification

Since \( 9^{1} \) simplifies to 9, that is the result of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that involves breaking down complex expressions into more manageable parts. This process often reveals underlying structures and patterns, helping solve or manipulate equations more easily. The expression in our exercise, \((\sqrt[3]{9})^3\), initially appears complex, but by simplifying it, we reveal a more straightforward solution.

To simplify an expression like this one, focus on the operations involved, here: taking a cube root and then raising to a power. By understanding and applying the rules of exponents and roots, the complex expression becomes easier to work with, demonstrating the power of algebraic techniques in simplifying problems.

Exponents

An exponent indicates how many times you multiply a base number by itself. In the expression \((9^{1/3})^3\), the exponent tells us to take the cube root (since \(1/3\) represents the cube root) and then raise it to the third power.

There are a few key rules to remember when working with exponents:

• Power of a Power: When you have an expression \((a^m)^n\), you multiply the exponents, resulting in \(a^{m \cdot n}\).
• Exponent of 1: Raising any number to the power of 1 leaves the number unchanged (e.g., \(9^1 = 9\)).
• Root as an Exponent: The nth root of a number can be expressed as a fractional exponent, such as \(\sqrt[3]{a} = a^{1/3}\).

Understanding these rules allows us to transition from \((9^{1/3})^3\) to \(9^1\), simplifying our expression.

Roots

Roots are the inverse operation of exponents. While exponents involve repeated multiplication, roots involve finding a number that, when raised to a given power, yields the original number. In the given expression, the cube root, \(\sqrt[3]{9}\), is initially calculated.

Understanding roots is crucial for simplifying expressions that involve fractional exponents. In our expression, \(\sqrt[3]{9}\) implies finding a number which, when cubed, returns 9. This understanding helps bridge the expression to other forms, such as using fractional exponents. This transformation of roots into a power form allows for easier simplification using standard exponent rules, ultimately reducing complexity and achieving a straightforward result of 9 after the simplification process.