Problem 84 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{1}{\sqrt[3]{4 m^{2}}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt[3]{2m}}{2m} \).

Step 1: Understand the Problem

The problem requires rationalizing the denominator of the fraction \( \frac{1}{\sqrt[3]{4m^2}} \). This means eliminating any radicals in the denominator.

Step 2: Identify the Cubic Root

The denominator is \( \sqrt[3]{4m^2} \), which is the cube root of \( 4m^2 \).

Step 3: Determine the Multiplier

To eliminate the cube root in the denominator, we can multiply by a radical that will transform the denominator into a perfect cube. Since the denominator is \( \sqrt[3]{4m^2} \), if we multiply by \( \sqrt[3]{2m} \), it gives us \( (\sqrt[3]{4m^2})(\sqrt[3]{2m}) = \sqrt[3]{8m^3} \).

Step 4: Rationalize the Denominator

Multiply both the numerator and the denominator by \( \sqrt[3]{2m} \): \[\frac{1}{\sqrt[3]{4m^2}} \times \frac{\sqrt[3]{2m}}{\sqrt[3]{2m}} = \frac{\sqrt[3]{2m}}{\sqrt[3]{8m^3}}.\]

Step 5: Simplify the Denominator

Since \( \sqrt[3]{8m^3} = 2m \), the denominator becomes a whole number: \[\frac{\sqrt[3]{2m}}{2m}.\]

Step 6: Simplify the Fraction

The fraction \( \frac{\sqrt[3]{2m}}{2m} \) is the rationalized form of the original expression, with the cube root completely removed from the denominator.

Key Concepts

Cube Roots

Understanding cube roots is pivotal when dealing with radical expressions, especially when looking to rationalize denominators. A cube root of a number or variable is a value that, when multiplied by itself twice more (or raised to the third power), gives the original number. For instance, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). In the exercise, the denominator \(\sqrt[3]{4m^2}\) involves finding a cube root of \(4m^2\). To simplify or rationalize, we seek a value that turns this into a perfect cube which eliminates the radical expression in the denominator once multiplied. This often involves determining which factors need to be added by multiplication, turning the expression into a perfect cube that is no longer under a radical.

Radical Expressions

Radical expressions, which include square roots, cube roots, or any root, can often look intimidating. They are expressions that involve roots and may require simplification for further calculations. For instance, a cube root such as \(\sqrt[3]{4m^2}\) denotes finding the number which, when raised to the power of three, results in \(4m^2\). This expression can often appear more complicated when it involves variables or coefficients, like \(m^2\), but the principles remain the same.

Since variables in radical expressions represent positive real numbers in this context, handling them often means:

• Identifying the base and power inside the root.
• Determining what multiplicative factor would turn the expression into a perfect power based on the index of the root (like 3 for cube roots).

By manipulating these expressions correctly, one can simplify or rationalize them to remove radicals from denominators resulting in a cleaner expression.

Fraction Simplification

Simplifying fractions is a common operation in algebra, especially when rationalizing denominators. The aim is to reduce a fraction to its simplest form, making further operations easier. This process often involves both rationalizing and simplifying.

When you encounter a fraction like \(\frac{1}{\sqrt[3]{4m^2}}\), you want to:

• Eliminate the radical in the denominator, as radicals can make calculations more complex.
• Multiply the numerator and the denominator by a strategic factor that will turn the denominator into a perfect cube.

After clearing the radical, you often achieve a fraction that needs additional simplification. For example, following the multiplication with \(\sqrt[3]{2m}\), the denominator turns into \(2m\), which simplifies the fraction. Therefore, rationalizing both removes complexity and sets the stage for using the expression in more advanced mathematical calculations.