Problem 86 - Algebra 2

Question

Rationalize the denominator of each expression. Assume that all variables are positive. \(\sqrt[5]{\frac{3 x^{3}}{2 y}}\)

Step-by-Step Solution

Answer

\(\frac{243 x^{15}}{32 y^{5}}\)

Step 1: Identify the Fraction Inside the Root

First, we need to identify the fraction which is inside the 5th root. In this case, the fraction inside the root is \( \frac{3 x^{3}}{2 y} \). Our task is to make the denominator rational, which means to get rid of the 'y' term in the denominator.

Step 2: Raise the Denominator to the Power of 5

We are given that all variables are positive, so we can safely raise the denominator to the power of 5 without affecting the rest of the expression. To retain the equality of the expression, we must also raise the numerator to the power of 5. This operation gives us: \( \sqrt[5]{\left(\frac{3 x^{3}}{2 y}\right)^{5}} \)

Step 3: Simplify the Expression

Next, simplify the expression by carrying out the power of 5 in the numerator and denominator. Reminder: \( (a/b)^n = a^n / b^n \). This operation simplifies to: \( \frac{3^{5} x^{15}}{2^{5} y^{5}} \)

Step 4: Simplify Further

We can simply each part of the fraction individually. The numerator simplifies to \( 243 x^{15} \) and the denominator simplifies to \( 32 y^{5} \). So the final expression is \( \frac{243 x^{15}}{32 y^{5}} \).

Key Concepts

Nth Roots

In mathematics, the concept of nth roots is central when exploring expressions involving radicals. An nth root of a number is a value that, when raised to the nth power, gives the original number. For instance, the square root (2nd root) and cube root (3rd root) are more commonly recognized examples, but any positive integer can serve as the "n" in "nth" root. In the given exercise, we are tasked with finding the 5th root, which means we want to find a number that, when raised to the power of 5, equals our original expression.

• The notation used for nth roots is \(\sqrt[n]{a}\), where "n" is the root and "a" is the number or expression under the root.
• Nth roots can often result in irrational numbers, meaning they cannot be expressed as simple fractions.

Understanding nth roots is crucial for rationalizing expressions, especially when simplifying or transforming radical denominators in rational expressions. It ensures we perform accurate manipulations when working towards simpler forms.

Rational Expressions

Rational expressions are fractions in which the numerator and the denominator are polynomials. They play a key role when simplifying and manipulating equations that include roots or radicals. In the expression \(\sqrt[5]{\frac{3x^3}{2y}}\), we see a classic example of a rational expression with both polynomials under a radical.

When working with these expressions, it's essential to understand:

• A rational expression can be written in the form \(\frac{P(x)}{Q(x)}\) where both "P" and "Q" are polynomials and \(Q(x) eq 0\).
• Simplifying rational expressions involves canceling common factors in the numerator and the denominator.

In this type of exercise, rationalizing the denominator is a typical goal. Simplifying the complex components of the rational expression often involves removing radicals from denominators to make calculations easier and the expression closer to a standard, rational form.

Simplifying Radicals

Simplifying radicals involves transforming a radical expression into its simplest form. This process typically aims to eliminate any radicals in the denominator, resulting in a more straightforward mathematical expression. In our exercise, we simplify the expression by dealing with the 5th root and bringing it into a standard rational format.

Key steps in simplifying radicals:

• Convert the radical to a different form using powers and exponents, such as \((a/b)^n\).
• Eliminate the radicals by raising the denominator to a power that transforms radical terms into rational terms, making the denominator a whole number.

Understanding how to handle and manipulate radicals can make complex expressions easier to understand and calculate, reducing errors and increasing efficiency.

Exponents and Powers

Exponents and powers are fundamental concepts in simplifying mathematical expressions involving radicals and rationalizing denominators. They provide a means to manipulate expressions efficiently by spacing out calculations into a more manageable form.

In exponents, the base number is multiplied by itself a number of times indicated by the exponent. For instance, in our example \((\frac{3 x^3}{2 y})^5\), the entire expression is raised to the power of 5. Here's what is essential to understand:

• The expression \((a/b)^n\) translates to \(a^n/b^n\).
• Exponent rules simplify expressions and are powerful in managing large numbers, especially in multiplication or division.
• Consistency in using these rules is crucial for accurate results, as seen in our exercise where both the numerator and the denominator were individually raised to the 5th power.

Exponents serve to simplify not only radicals but all forms of expressions, transforming them into more usable, logical terms.