Problem 70 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{3}{\sqrt[3]{3}}\)

Step-by-Step Solution

Answer

The simplest radical form is \( \sqrt[3]{9} \).

Step 1: Identify the Radical

The expression given is \( \frac{3}{\sqrt[3]{3}} \). Here, the denominator contains a cube root, which is a radical. Our task is to simplify this radical.

Step 2: Rationalize the Denominator

To simplify \( \frac{3}{\sqrt[3]{3}} \), we must eliminate the cube root from the denominator. We do this by multiplying both the numerator and the denominator by \( \sqrt[3]{9} \), which is the cube root of \( 3^2 \). This action, \( \frac{3}{\sqrt[3]{3}} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{9}} \), helps in removing the radical from the denominator.

Step 3: Simplify the Expression

After multiplying, the expression becomes \( \frac{3 \cdot \sqrt[3]{9}}{\sqrt[3]{27}} \). Recognize that \( \sqrt[3]{27} \) is equivalent to \( 3 \), since \( 3^3 = 27 \). So the expression simplifies further to \( \frac{3 \cdot \sqrt[3]{9}}{3} \).

Step 4: Cancel Common Terms

The \( 3 \) in the numerator and the denominator can be canceled out, simplifying the expression to \( \sqrt[3]{9} \).

Step 5: Final Simplified Form

The final simplified form of the original expression \( \frac{3}{\sqrt[3]{3}} \) is \( \sqrt[3]{9} \).

Key Concepts

Rationalizing the Denominator

When you encounter a mathematical expression with a radical in the denominator, especially one involving cube roots, the goal is to "rationalize" the denominator. This means making the denominator a rational number—it shouldn’t have any radicals like square roots or cube roots. In our exercise, we started with the expression \( \frac{3}{\sqrt[3]{3}} \). Here, the denominator is \( \sqrt[3]{3} \), a cube root. To eliminate this cube root, we multiply the entire fraction by a value that will rid the denominator of the radical.

• Identify the cube root or power needed to rationalize. In this case, it was \( \sqrt[3]{9} \) because \( (\sqrt[3]{3})^2 =\sqrt[3]{9} \).
• Multiply both the numerator and the denominator by this value: \( \frac{3}{\sqrt[3]{3}} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{9}} \).
• This process transforms the denominator \( \sqrt[3]{3} \times \sqrt[3]{9} = \sqrt[3]{27} \), simplifying to \( 3 \).

Rationalizing the denominator helps in simplifying the expression to make it easier to work with in further calculations.

Cube Roots

Cube roots are similar to square roots, but instead of finding a number that when squared gives the original number, cube roots find a number that when cubed gives the original. The cube root of a number \( x \) is written as \( \sqrt[3]{x} \), and it solves for \( y \) such that \( y^3 = x \).In the given problem, the cube root \( \sqrt[3]{3} \) in the denominator needed simplifying. To do this, we look for a number which, when multiplied by \( \sqrt[3]{3} \), yields a perfect cube.

• The value \( \sqrt[3]{9} \) was chosen since multiplying it by \( \sqrt[3]{3} \) results in \( \sqrt[3]{27} \).
• The result, 27, is a perfect cube because \( 3^3 = 27 \).

By converting the denominator into a perfect cube, it simplifies to an integer, allowing the expression to be simplified further.

Algebraic Expressions

Working with algebraic expressions requires manipulating various mathematical elements such as numbers, variables, and radicals to simplify them. In our exercise, the expression involved simplifying a fraction with a radical term in the denominator.Algebraic expressions often involve operations including addition, subtraction, multiplication, and division. When simplifying, the goal is to reduce the expression to its simplest form while maintaining equivalent value.

• Multiplying the expression \( \frac{3}{\sqrt[3]{3}} \) by \( \frac{\sqrt[3]{9}}{\sqrt[3]{9}} \) changed its form but not its value as both the numerator and denominator were altered by the same quantity.
• Recognize common terms or factors that can be cancelled out—in this case, the 3s that appear in the numerator and the denominator once \( \sqrt[3]{27} = 3 \).

This approach highlights how understanding and manipulating algebraic expressions can make complex problems much more manageable.