Rationalizing Denominators Simplifying Cube Root Expressions

In the realm of mathematics, simplifying expressions involving radicals often requires a technique called rationalizing the denominator. This process eliminates radicals from the denominator of a fraction, making it easier to work with and compare expressions. This article delves into the intricacies of rationalizing denominators, particularly focusing on cube roots and variables, and provides a step-by-step approach to simplifying such expressions.

Understanding Rationalizing the Denominator

Rationalizing the denominator is a crucial skill in algebra and beyond, ensuring that expressions are presented in their simplest and most conventional form. The primary goal is to remove any radical expressions from the denominator of a fraction. This is achieved by multiplying both the numerator and denominator by a suitable expression that will eliminate the radical in the denominator. The key principle behind this technique lies in the property that multiplying a radical by itself (or a power of itself) can result in a rational number. For instance, multiplying a square root by itself eliminates the radical, while multiplying a cube root by its square achieves the same effect. In the case of the expression $\frac{\sqrt[3]{15 b^4}}{\sqrt[3]{b}}$, we are dealing with cube roots, which means we need to transform the denominator into a perfect cube to eliminate the radical. This involves identifying the factor needed to complete the cube under the radical in the denominator and then multiplying both the numerator and denominator by that factor. The process not only simplifies the expression but also adheres to mathematical conventions for presenting radical expressions in their most reduced form. Rationalizing the denominator is not merely an aesthetic preference; it also facilitates further calculations and comparisons of expressions, making it an essential tool in mathematical manipulations.

Step-by-Step Solution for the Given Expression

To effectively rationalize the denominator of the expression $\frac{\sqrt[3]{15 b^4}}{\sqrt[3]{b}}$, we need to carefully follow a series of steps that will eliminate the radical in the denominator while maintaining the integrity of the expression. The process begins with a thorough analysis of the denominator, identifying the factor needed to transform it into a perfect cube. In this case, the denominator is $\sqrt[3]{b}$, which means we need to multiply it by a factor that will result in $b^3$ under the cube root. The logical choice is $\sqrt[3]{b^2}$, as multiplying $\sqrt[3]{b}$ by $\sqrt[3]{b^2}$ yields $\sqrt[3]{b^3}$, which simplifies to $b$. Once we have identified the appropriate factor, the next step involves multiplying both the numerator and the denominator by this factor. This ensures that we are essentially multiplying the expression by 1, preserving its value while altering its form. After multiplying, we simplify the resulting expression by combining the radicals in the numerator and denominator and then extracting any perfect cube roots. This may involve breaking down the radicands into their prime factors and identifying groups of three, which can be taken out of the cube root. The final step is to reduce the fraction, if possible, by canceling out any common factors between the numerator and the denominator. This may involve simplifying both the numerical coefficients and the variable terms, ensuring that the expression is presented in its most concise and simplified form. By adhering to these steps, we can effectively rationalize the denominator and simplify radical expressions, making them easier to interpret and use in further calculations.

1. Identify the Factor Needed to Rationalize the Denominator

The initial step in rationalizing the denominator of the given expression, $\frac{\sqrt[3]{15 b^4}}{\sqrt[3]{b}}$, is to pinpoint the exact factor that will eliminate the cube root from the denominator. The denominator in question is $\sqrt[3]{b}$, which means we are dealing with a cube root. To rationalize this denominator, our objective is to transform the expression under the cube root into a perfect cube. A perfect cube is an expression that can be written as something raised to the power of three. In this scenario, we want to transform $b$ into $b^3$. To achieve this, we need to multiply $b$ by $b^2$, since $b \cdot b^2 = b^3$. Therefore, the factor we need to multiply the denominator by is $\sqrt[3]{b^2}$. This is because $\sqrt[3]{b} \cdot \sqrt[3]{b^2} = \sqrt[3]{b^3}$, and the cube root of $b^3$ is simply $b$, which is a rational expression. Identifying this factor is crucial as it forms the basis for the subsequent steps in the rationalization process. It allows us to manipulate the expression in a way that removes the radical from the denominator without changing the overall value of the expression. This step requires a clear understanding of the properties of radicals and exponents, and it sets the stage for the simplification that follows.

2. Multiply the Numerator and Denominator by the Factor

Once we've identified the necessary factor to rationalize the denominator—in this case, $\sqrt[3]{b^2}$—the next step is to multiply both the numerator and the denominator of the original expression, $\frac{\sqrt[3]{15 b^4}}{\sqrt[3]{b}}$, by this factor. This process is crucial because it allows us to change the form of the expression without altering its value. Multiplying both the numerator and denominator by the same factor is equivalent to multiplying the entire expression by 1, which maintains its original value while enabling us to eliminate the radical in the denominator. So, we multiply $\frac{\sqrt[3]{15 b^4}}{\sqrt[3]{b}}$ by $\frac{\sqrt[3]{b^2}}{\sqrt[3]{b^2}}$. This gives us a new expression: $\frac{\sqrt[3]{15 b^4} \cdot \sqrt[3]{b^2}}{\sqrt[3]{b} \cdot \sqrt[3]{b^2}}$. This step is a fundamental application of the principle that multiplying the numerator and denominator of a fraction by the same non-zero quantity does not change the fraction's value. It sets the stage for the simplification process, where we combine the radicals and then extract any perfect cubes. The careful execution of this step is essential for the successful rationalization of the denominator and the overall simplification of the expression. By multiplying by the appropriate factor, we pave the way for removing the radical from the denominator and expressing the result in its simplest form.

3. Simplify the Expression

After multiplying the numerator and the denominator by the factor $\sqrt[3]{b^2}$, we arrive at the expression $\frac{\sqrt[3]{15 b^4} \cdot \sqrt[3]{b^2}}{\sqrt[3]{b} \cdot \sqrt[3]{b^2}}$. The next crucial step is to simplify this expression. This involves combining the radicals in both the numerator and the denominator and then extracting any perfect cubes. Starting with the numerator, we have $\sqrt[3]{15 b^4} \cdot \sqrt[3]{b^2}$. According to the properties of radicals, we can combine these into a single cube root: $\sqrt[3]{15 b^4 \cdot b^2}$, which simplifies to $\sqrt[3]{15 b^6}$. Similarly, in the denominator, we have $\sqrt[3]{b} \cdot \sqrt[3]{b^2}$, which combines to $\sqrt[3]{b \cdot b^2}$, simplifying to $\sqrt[3]{b^3}$. Now, our expression looks like this: $\frac{\sqrt[3]{15 b^6}}{\sqrt[3]{b^3}}$. Next, we simplify the radicals by extracting perfect cubes. In the numerator, $b^6$ is a perfect cube because $b^6 = (b^2)^3$. Thus, $\sqrt[3]{15 b^6}$ can be written as $\sqrt[3]{15} \cdot \sqrt[3]{b^6}$, which simplifies to $b^2 \sqrt[3]{15}$. In the denominator, $\sqrt[3]{b^3}$ simplifies to $b$. So, our expression now becomes $\frac{b^2 \sqrt[3]{15}}{b}$. The final part of the simplification involves reducing the fraction by canceling out any common factors between the numerator and the denominator. In this case, we can divide both the numerator and the denominator by $b$, which leaves us with the simplified expression $b \sqrt[3]{15}$. This simplification process is a critical part of rationalizing the denominator, as it ensures that the expression is presented in its most concise and understandable form.

4. Final Answer

Following the steps of rationalizing the denominator and simplifying the expression, we have successfully transformed the original expression, $\frac{\sqrt[3]{15 b^4}}{\sqrt[3]{b}}$, into its simplest form. After identifying the factor needed to rationalize the denominator, multiplying both the numerator and the denominator by that factor, and then simplifying the resulting expression, we arrive at the final answer: $b \sqrt[3]{15}$. This final form is not only free of radicals in the denominator but is also expressed in its most reduced form, adhering to the conventions of mathematical notation. The process involved combining radicals, extracting perfect cubes, and reducing fractions, all of which are fundamental skills in algebraic manipulation. The result, $b \sqrt[3]{15}$, is a clear and concise representation of the original expression, making it easier to work with in further calculations or comparisons. This demonstrates the power of rationalizing the denominator as a technique for simplifying radical expressions and presenting them in a standardized and easily interpretable manner.

Conclusion

In conclusion, rationalizing the denominator is a vital technique in simplifying radical expressions, ensuring they are presented in their most accessible and conventional form. By systematically identifying the necessary factors, multiplying both the numerator and denominator, and simplifying the result, we can effectively eliminate radicals from the denominator. In the specific case of $\frac{\sqrt[3]{15 b^4}}{\sqrt[3]{b}}$, we successfully transformed the expression into $b \sqrt[3]{15}$, demonstrating the power and utility of this method. This skill is not only essential for algebraic manipulations but also enhances our understanding of mathematical expressions and their properties. Mastering this technique allows for more efficient and accurate problem-solving in various mathematical contexts.