Problem 117 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \frac{\sqrt[3]{x}}{\sqrt[3]{9}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\sqrt[3]{81x}}{9} \).

Step 1: Identifying the Expression

The given expression is \( \frac{\sqrt[3]{x}}{\sqrt[3]{9}} \). We need to simplify this expression by performing operations on radicals.

Step 2: Rationalizing the Denominator

To eliminate the cube root in the denominator, we will multiply both the numerator and the denominator by \( \sqrt[3]{9^2} \). This will give us \( \frac{\sqrt[3]{x} \cdot \sqrt[3]{9^2}}{\sqrt[3]{9} \cdot \sqrt[3]{9^2}} \), which simplifies to \( \frac{\sqrt[3]{x \cdot 9^2}}{9} \).

Step 3: Simplifying the Radical Expression

Calculate \( 9^2 = 81 \). Substitute back to get \( \frac{\sqrt[3]{81x}}{9} \). This is the simplified form, as the cube root is now only in the numerator and the denominator is rational.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves the process of eliminating any radicals—such as square roots or cube roots—found in the denominator of a fraction. This can make the expression simpler to work with and is generally preferred in mathematical solutions.

• For the given expression, \( \frac{\sqrt[3]{x}}{\sqrt[3]{9}} \), we had a radical in the denominator, \( \sqrt[3]{9} \).
• To rationalize this, we multiply both the numerator and the denominator by the radical expression needed to make the denominator a whole number.
• In this case, multiplying both numerator and denominator by \( \sqrt[3]{9^2} \) helps achieve this goal, as \( \sqrt[3]{9} \times \sqrt[3]{9^2} = 9 \), which is a rational number.

This step of rationalizing makes the expression easier to handle and interpret.

Cube Roots

Cube roots are very similar to square roots, but instead of finding what number multiplied by itself twice equals a given number, a cube root finds what number multiplied by itself three times gives the original number.

• The cube root symbol is \( \sqrt[3]{x} \), where \( x \) is the radicand, or the number under the cube root symbol.
• If you have \( \sqrt[3]{x}^3 = x \), this helps show that raising a cube root to the power of 3 returns the original radicand.
• In our exercise, we are working with cube roots in both the numerator and denominator; \( \sqrt[3]{x} \) represents the cube root of \( x \), and \( \sqrt[3]{9} \) is the cube root of \( 9 \).

Handling cube roots efficiently is key in simplifying expressions like ours, making sure to combine or reduce wherever possible.

Radical Operations

Radical operations can involve addition, subtraction, multiplication, or division of radical expressions. Similar to other mathematical operations, specific rules and properties help in simplifying these expressions.

• Add or subtract radicals only when they have the same type and radicand. For example, \( 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3} \) is valid because they are like terms.
• To multiply or divide radicals, use the property \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \). This helps in combining radicands under the same radical.
• For division, simplify by reducing the radicals, like we've done by multiplying \( \sqrt[3]{x} \) and \( \sqrt[3]{9^2} \).

In our expression \( \frac{\sqrt[3]{x} \cdot \sqrt[3]{9^2}}{9} \), we ultimately used division and multiplication of radicals to arrive at a much easier form to understand, by ensuring we simplify every possible radical component.