Problem 17 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[4]{\frac{x^{3}}{16}} $$

Step-by-Step Solution

Answer

\( \frac{x^{3/4}}{2} \) is the simplified form.

Step 1: Rewriting the Expression

First, rewrite the fourth root expression \( \sqrt[4]{\frac{x^{3}}{16}} \) in terms of exponentiation. Recall that \( \sqrt[4]{a} = a^{1/4} \). Thus, \( \sqrt[4]{\frac{x^{3}}{16}} = \left( \frac{x^{3}}{16} \right)^{1/4} \).

Step 2: Applying the Power of a Quotient Rule

Apply the power of a quotient rule \( (\frac{a}{b})^{n} = \frac{a^n}{b^n} \) to the expression \( \left( \frac{x^{3}}{16} \right)^{1/4} \). This gives us \( \frac{(x^{3})^{1/4}}{16^{1/4}} \).

Step 3: Simplifying the Numerator

Simplify \( (x^3)^{1/4} \) using the rule \( (a^m)^n = a^{m } \). So, \( (x^3)^{1/4} = x^{3/4} \).

Step 4: Simplifying the Denominator

Simplify \( 16^{1/4} \). Since \( 16 = 2^4 \), then \( 16^{1/4} = (2^4)^{1/4} = 2^{4 \times \frac{1}{4}} = 2^1 = 2 \).

Step 5: Writing the Final Simplified Expression

Combine the simplified numerator and denominator to express the simplified form of the original expression as \( \frac{x^{3/4}}{2} \).

Key Concepts

Exponentiation

Exponentiation is an essential mathematical operation involving numbers. It indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \( x^3 \), \( x \) is multiplied three times. The small number 3 is called the exponent, which signifies how many times the multiplication occurs. Exponentiation can transform different expressions, making them easier to handle, especially with roots or fractional exponents.
In dealing with roots like the fourth root in the exercise, exponentiation allows us to rewrite roots as fractions. For instance, the fourth root of \( x \), written as \( \sqrt[4]{x} \), can be expressed in terms of exponentiation as \( x^{1/4} \). Understanding this conversion is key in simplifying complex expressions using other rules like the Power of a Quotient Rule.

Power of a Quotient Rule

The Power of a Quotient Rule is a valuable property in algebra, which simplifies expressions that involve exponents. This rule states that when you have a quotient raised to an exponent, like \( \left( \frac{a}{b} \right)^n \), you can distribute the exponent to both the numerator and the denominator. This means it can be written as \( \frac{a^n}{b^n} \).
Applying this rule in our exercise, we take the expression \( \left( \frac{x^3}{16} \right)^{1/4} \) and transform it to \( \frac{(x^3)^{1/4}}{16^{1/4}} \). This step simplifies dealing with the separate components, reducing the difficulty of working with complex radical expressions.
It's crucial to grasp this rule as it eases the process of solving equations and simplifies expressions effectively, making complex calculations more manageable. Whether dealing with numbers or variables, the Power of a Quotient Rule helps break down and solve mathematical problems.

Simplification Steps

Simplification in mathematics involves breaking down expressions into simpler or more manageable forms. Let's look at the approach to simplifying the expression from the exercise step by step.

• First, recognize that \( \frac{x^3}{16} \) under a fourth root can be written as \( \left( \frac{x^3}{16} \right)^{1/4} \).
• Then, apply the Power of a Quotient Rule: separate the numerator and the denominator, resulting in \( \frac{(x^3)^{1/4}}{16^{1/4}} \).
• To simplify the numerator \( (x^3)^{1/4} \), use the rule \((a^m)^n = a^{m \times n} \), which gives us \( x^{3/4} \).
• For the denominator, simplify \( 16^{1/4} \) knowing that \( 16 = 2^4 \); thus, \( 16^{1/4} = 2 \).

Combining the simplified parts, the expression now reads \( \frac{x^{3/4}}{2} \). Each step is logical and follows fundamental algebraic rules, demonstrating the power of techniques like exponentiation and the Power of a Quotient Rule in mathematical simplification.

Radical Expression Simplification

Radical Expression Simplification is the process of transforming expressions containing roots into simpler forms. Involving both radicals and exponents, this process enhances understanding and application in more complex problems. Since radicals can sometimes complicate calculations, expressing them using fractional exponents can make them more approachable.
The original problem \( \sqrt[4]{\frac{x^3}{16}} \), is a radical expression that benefits from simplification. By rewriting it as \( \left( \frac{x^3}{16} \right)^{1/4} \) using exponentiation, and applying rules like the Power of a Quotient, we manage each part separately. Simplifying the numerator and the denominator step-by-step provides a clear solution: \( \frac{x^{3/4}}{2} \).
This method proves particularly useful for handling complex expressions, reducing them to a simpler, more interpretable form without losing the essence of the original mathematical problem.