Problem 30 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{9 y}{\sqrt[4]{4 y^{9}}}\)

Step-by-Step Solution

Answer

\( \frac{9\sqrt{2}}{2y^{5/4}} \)

Step 1: Simplify the Radical

Given the expression \( \sqrt[4]{4y^9} \), start by simplifying the fourth root. Rewrite it as \( (4y^9)^{1/4} \). Apply the property \( (ab)^n = a^n b^n \), we get \( 4^{1/4} \cdot (y^9)^{1/4} \). This becomes \( 2^{1/2} \cdot y^{9/4} \), which simplifies to \( \sqrt{2} \cdot y^{9/4} \).

Step 2: Rationalize the Denominator

To rationalize the denominator \( \sqrt{2} \cdot y^{9/4} \), multiply the numerator and denominator by \( \sqrt{2} \cdot y^{7/4} \). This choice makes the powers of \( y \) become a whole number. The denominator becomes \( (\sqrt{2})^2 \cdot y^{9/4+7/4} = 2 \cdot y^{4} = 2y^{4} \). The numerator becomes \( 9y \cdot \sqrt{2} \cdot y^{7/4} = 9\sqrt{2} y^{1+7/4} = 9\sqrt{2} y^{11/4} \).

Step 3: Simplify the Expression

Simplify the expression \( \frac{9\sqrt{2} y^{11/4}}{2y^{4}} \). Divide each term in the numerator by the denominator: \( \frac{9\sqrt{2}}{2} \cdot \frac{y^{11/4}}{y^4} = \frac{9\sqrt{2}}{2} \cdot y^{11/4 - 4} = \frac{9\sqrt{2}}{2} \cdot y^{-5/4} \).

Step 4: Final Simplification: Rewriting with Positive Exponents

Rewrite \( y^{-5/4} \) as \( \frac{1}{y^{5/4}} \) to express it with a positive exponent. So the expression simplifies to \( \frac{9\sqrt{2}}{2} \cdot \frac{1}{y^{5/4}} = \frac{9\sqrt{2}}{2y^{5/4}} \).

Key Concepts

Exponents Simplification

In mathematics, exponents indicate how many times a number or variable is multiplied by itself. Simplifying exponents involves making these expressions as straightforward as possible. This is essential for solving algebraic problems effectively. When you encounter expressions with exponents, it's helpful to remember these basic rules:

• Product of Powers: When multiplying two expressions with the same base, add their exponents. For example, \(x^a \cdot x^b = x^{a+b}\).
• Power of a Power: When raising a power to another power, multiply the exponents. For instance, \((x^a)^b = x^{ab}\).
• Power of a Product: Distribute the exponent to every factor within the parentheses. For example, \((xy)^n = x^n y^n\).

Understanding these rules can greatly simplify the process of rationalizing the denominator in an expression. It's crucial to simplify exponents to maintain mathematical accuracy and clarity, especially when dealing with more complex algebraic problems.

Radicals

Radicals refer to expressions that include roots, such as square roots, cube roots, or fourth roots. Simplifying radicals is a common task in algebra that makes working with such expressions more manageable. For the expression \(\sqrt[4]{4y^9}\), it represents the fourth root of \(4y^9\). To simplify such radical expressions, there's a useful strategy involving converting them into exponents. Here’s how you can go about doing this:

• Convert to Exponents: Rewrite radicals using fractional exponents, such as \(\sqrt[n]{x} = x^{1/n}\). This makes applying exponential rules easier.
• Apply Exponent Rules: Use the exponent rules to simplify the expression further. For instance, break it down by distributing the root across each component, like converting \(\sqrt[4]{4y^9} = 4^{1/4} \cdot y^{9/4}\).
• Simplify Each Part: Use known values (like converting \(4^{1/4} = \sqrt{2}\)) and simplify the variable part using appropriate exponent algebra.

Simplifying radicals is essential to solving equations efficiently, especially in more complex algebraic contexts.

Expressions with Positive Exponents

Expressing exponents positively is a vital aspect of algebra to ensure expressions are easy to read and compare. When exponents are negative, they can often be rewritten as positive exponents for more clarity. For instance, in expressions like \(y^{-5/4}\), rewriting it as a positive exponent involves finding its reciprocal:

• Rewrite as a Fraction: Negative exponents can be converted to positive by taking the reciprocal of the base with a positive exponent, such as \(y^{-a} = \frac{1}{y^a}\).
• Apply to Algebraic Expressions: When an expression includes a negative exponent, apply this rule to convert each part accordingly. In the given problem, this converts \(y^{-5/4}\) to \(\frac{1}{y^{5/4}}\).

This process helps in keeping expressions straightforward and prevents misunderstandings in calculations. Converting all parts of an algebraic expression to positive exponents improves both clarity and mathematical accuracy.