Problem 73 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[5]{\frac{5 x^{7} y^{2}}{8 x^{3}}}$$

Step-by-Step Solution

Answer

\(\frac{\sqrt[5]{320x^4y^2}}{2}\)

Step 1: Simplify the Fraction Inside the Radical

We start by simplifying the expression inside the fifth root. The original expression is \( \frac{5x^7 y^2}{8x^3} \). Divide the terms with similar bases: \( \frac{x^7}{x^3} = x^{7-3} = x^4 \). Thus, the expression simplifies to \( \frac{5x^4 y^2}{8} \).

Step 2: Apply the Fifth Root to Each Term

Next, apply the property of radicals where \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \). Our expression becomes \( \frac{\sqrt[5]{5x^4y^2}}{\sqrt[5]{8}} \).

Step 3: Simplify the Fifth Root in the Numerator

Focus on the numerator: \( \sqrt[5]{5x^4y^2} \). Since there are no perfect fifth powers in 5, \(x^4\), or \(y^2\), leave this radical as is: \( \sqrt[5]{5x^4y^2} \).

Step 4: Simplify the Fifth Root of the Constant in the Denominator

Now look at the denominator: \( \sqrt[5]{8} \). Since \(8 = 2^3\) and 3 is less than 5, it is not a perfect fifth power, so this stays as \( \sqrt[5]{8} \).

Step 5: Rationalize the Denominator

To rationalize \( \sqrt[5]{8} \), multiply the numerator and the denominator by \( \sqrt[5]{8^2} = \sqrt[5]{64} \) so the denominator exponent sums to 5. The expression becomes \( \frac{\sqrt[5]{5x^4y^2} \cdot \sqrt[5]{64}}{8} \).

Step 6: Combine Terms and Simplify the Result

The numerator becomes \( \sqrt[5]{320x^4y^2} \). The denominator is now \( 2 \), because \( \sqrt[5]{8^3} = \sqrt[5]{512} = 8 \). The simplified expression is \( \frac{\sqrt[5]{320x^4y^2}}{2} \).

Key Concepts

Simplifying Radicals

Simplifying radicals means making root expressions easier to manage while keeping their value the same. When we simplify, we aim to resolve the expression into a form that's easier to understand or work with. Here's a quick guide:

• Identify any perfect powers: Look for numbers or variables raised to powers that match the index of the radical (the little number outside the root symbol). These can be taken out of the radical.
• Prime Factorization: Break down numbers into their prime factors to check for any common powers matching the index.
• Cancel Powers: If possible, simplify the expression by cancelling out the powers inside the radical that match the index.

For example, when simplifying \[\sqrt[5]{x^5} \] we recognize that because 5 and 5 match, we can simplify it as \(x\) without the radical. Simplifying radicals simplifies calculations and expressions, much like tidying up a room.

nth Roots

The concept of nth roots extends the idea of square roots to any integer 'n', making it a generalized form of roots. An nth root of a number is another number which, when multiplied by itself 'n' times, results in the original number. Often seen as:

• Radical Notation: \(\sqrt[n]{a}\) represents the nth root of \(a\).
• Understanding the Index: The 'n' is known as the index or degree of the root, indicating how many times the root factor should multiply to reach the original number.
• Imaginary and Complex Roots: If \(n\) is even and \(a\) is negative, we talk about imaginary roots because no real number multiplied by itself an even number of times will result in a negative product.

For instance, in the problem, \(\sqrt[5]{8}\), the '5' is the index indicating a fifth root. It helps in breaking down numbers and expressions into simpler forms, providing a method to handle complex expressions in mathematics more effectively.

Exponent Rules

Exponent rules provide the foundation for manipulating expressions involving powers. These rules are essential when dealing with problems that have repeated multiplication of the same base. Here are key rules to remember:

• Product of Powers: \(x^a \cdot x^b = x^{a+b}\)
• Quotient of Powers: \(\frac{x^a}{x^b} = x^{a-b}\)
• Power of a Power: \((x^a)^b = x^{a \cdot b}\)
• Power of a Product: \((xy)^a = x^a y^a\)

Using these rules simplifies complex expressions by reducing the number of operations needed. In our exercise, the quotient of powers rule allowed us to simplify \(\frac{x^7}{x^3} = x^{4}\), thereby clarifying the expression quickly and effectively.

Fractions in Mathematics

Fractions represent parts of a whole and are composed of a numerator and a denominator. They are essential in mathematical operations, especially when working with division, proportional relationships, and radical expressions. Key concepts to understand about fractions include:

• Simplifying Fractions: Dividing both the numerator and the denominator by their greatest common divisor to make the fraction as simple as possible.
• Equivalent Fractions: Different fractions which represent the same value, like \(\frac{1}{2}\) and \(\frac{2}{4}\).
• Rationalizing: Involves making the denominator of a fraction a rational number, often by eliminating radicals.

In our problem, rationalizing involves multiplying the fraction by a form of 1, such as \(\frac{\sqrt[5]{64}}{\sqrt[5]{64}}\), to clear radicals from the denominator, thus ensuring the expression is easier to interpret and work with. Understanding how to handle fractions aids in making complex expressions more approachable.