Problem 83 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{19}{\sqrt[3]{5 c^{2}}} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{19 \sqrt[3]{25c^4}}{5c} \).

Step 1: Identify the Issue

We need to rationalize the denominator of the expression \( \frac{19}{\sqrt[3]{5c^2}} \). The denominator contains a cube root, which we need to eliminate by multiplying by a value that will convert it into a whole number.

Step 2: Setup the Multiplication Factor

To remove the cube root, multiply the numerator and the denominator by \( \sqrt[3]{(5c^2)^2} \). This is because multiplying reduces the cube root and results in \( \sqrt[3]{(5c^2)^3} \) in the denominator.

Step 3: Multiply Numerator and Denominator

Multiply both the numerator and the denominator by \( \sqrt[3]{(5c^2)^2} \). This gives us: \[ \frac{19 \times \sqrt[3]{(5c^2)^2}}{\sqrt[3]{5c^2} \times \sqrt[3]{(5c^2)^2}} = \frac{19 \sqrt[3]{25c^4}}{\sqrt[3]{(5c^2)^3}} \].

Step 4: Simplify the Denominator

Simplify the denominator, \( \sqrt[3]{(5c^2)^3} \), to 5c because raising a cube root to the third power returns the base.

Step 5: Expression Simplification

The expression now becomes \( \frac{19 \sqrt[3]{25c^4}}{5c} \). This represents the final solution with a rationalized denominator with no cube roots.

Key Concepts

Cube Root

The cube root is an important concept where you find a number that, when multiplied by itself three times, equals the original number. In our given problem, the cube root of a number, specifically \( \sqrt[3]{5c^2} \), is present in the denominator of a rational expression. Cube roots can be tricky to handle in expressions because they involve fractional powers.

To rationalize the denominator containing a cube root, the goal is to eliminate it by multiplying with an appropriate factor. In this case, we observe that multiplying the cube root of a number by the same base raised to two more powers (e.g., \( \sqrt[3]{(5c^2)^2} \)) results in eliminating the root. This is because

• \( (5c^2) \times (5c^2)^2 = (5c^2)^3 \)
• The cube root of \( (5c^2)^3 \) simplifies to \( 5c^2 \)

Bear in mind that all variables involved are positive real numbers, which means the operations are valid within real numbers, and there is no involvement of complex numbers here.

Rational Expression

An expression involving a fraction, where both the numerator and the denominator are polynomials, is known as a rational expression. In our case, the expression \( \frac{19}{\sqrt[3]{5c^2}} \) is a rational expression with a denominator that needs rationalization because it contains a cube root.

When handling rational expressions, especially with roots, the focus is to simplify them so that no roots remain in the denominator. This is a key practice because arithmetic operations are generally easier with whole numbers or simpler terms. To manage and simplify such expressions:

• Ensure the denominator is devoid of roots or irrational numbers.
• Multiply both the numerator and the denominator by an appropriate radical term that neutralizes the root.
• Simplify the resulting expression as much as possible.

This ensures the final expression is in its simplest, easily interpretable form, thus making further computations simpler.

Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form without altering its value. In our problem, we simplify the expression \( \frac{19\sqrt[3]{25c^4}}{5c} \) after rationalizing its denominator.

To achieve this, follow steps to ensure:

• Combine like terms where possible.
• Simplify any radical expressions.
• Cancel any common factors present in the numerator and the denominator.

For example, by rationalizing we ensure the cube root is removed, leaving us with just simple multiplication in the numerator and division by a monomial in the denominator. These operations are typical for expressing the result in a form that is easier to compute, analyze, or integrate further into more complex problems. Thus, proficiency in simplification tactics is critical for handling algebraic expressions efficiently and accurately.