Problem 22 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[4]{\frac{a^{3}}{81}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{a^{3/4}}{3} \).

Step 1: Express as Exponents

First, express the given fourth root in terms of exponents. The fourth root of a number can be expressed as raising that number to the power of \( \frac{1}{4} \). Thus, \[ \sqrt[4]{\frac{a^3}{81}} = \left(\frac{a^3}{81}\right)^{\frac{1}{4}}. \]

Step 2: Apply the Power of a Quotient Property

The power of a quotient property states \( \left(\frac{b}{c}\right)^n = \frac{b^n}{c^n} \). Apply this property:\[ \left(\frac{a^3}{81}\right)^{\frac{1}{4}} = \frac{(a^3)^{\frac{1}{4}}}{(81)^{\frac{1}{4}}}. \]

Step 3: Simplify the Numerator

Simplify \((a^3)^{\frac{1}{4}}\) using the power of a power property, which states \((b^m)^n = b^{m \times n}\): \[ (a^3)^{\frac{1}{4}} = a^{3 \times \frac{1}{4}} = a^{\frac{3}{4}}. \]

Step 4: Simplify the Denominator

Express 81 as a base raised to a power. Note that 81 is \(3^4\). Thus:\[ (81)^{\frac{1}{4}} = (3^4)^{\frac{1}{4}}. \]Using the power of a power property, \[ (3^4)^{\frac{1}{4}} = 3^{4 \times \frac{1}{4}} = 3^1 = 3. \]

Step 5: Combine the Simplified Parts

Combine the simplified numerator and denominator:\[ \frac{a^{\frac{3}{4}}}{3}. \] This is our simplified expression.

Key Concepts

Power of a Quotient Property

The power of a quotient property is a valuable tool when simplifying expressions involving fractions raised to powers. It states that if you have a fraction \( \frac{b}{c} \) raised to an exponent \((n)\), it can be expressed as \((\frac{b^n}{c^n})\).
This property allows us to separate the numerator and the denominator and treat their exponents individually.
For example, if you have \((\frac{a^3}{81})^{\frac{1}{4}}\), you can use this property to simplify it as \((\frac{a^3}{81})^{\frac{1}{4}} = \frac{(a^3)^{\frac{1}{4}}}{(81)^{\frac{1}{4}}}\).

• Simplifying the numerator and denominator separately makes the process easier.
• It leverages both the power and root properties.

Furthermore, this property becomes extremely useful when dealing back with complex fractions or radicals. Handling the components separately often helps students manage the complexity one step at a time.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number. In exponential notation, \((b^m)\), \(b\) is the base and \(m\) is the exponent, representing that \(b\) is multiplied by itself \(m\) times.
Understanding exponents allows us to simplify expressions and solve equations more efficiently.

• For instance, the expression \((a^3)^{\frac{1}{4}}\) uses the power of a power property, which lets you multiply the exponents: \(a^{3 \times \frac{1}{4}} = a^{\frac{3}{4}}\).
• This property says that \((b^m)^n = b^{m \times n}\).

These properties reduce complex expressions down to a simpler form, making your calculations more manageable and less prone to mistakes. Understanding and applying these rules correctly is essential for tackling advanced math problems.

Simplifying Expressions

Simplifying expressions is all about making a complex expression easier to understand and work with by reducing it to its simplest form. This means following a series of rules and properties that allow us to clear up the clutter.

• When simplifying fractions with exponents, apply your knowledge of exponent rules to break down the terms: \(\frac{a^{\frac{3}{4}}}{3}\) becomes much easier to manage.
• Combining all elements ultimately gives you the form you see in the step-by-step solution: a simpler and tidier expression.

Simplification makes it easier to see the essence of the problem and spot potential errors or avenues to the solution. It also helps prepare complex expressions for further operations, like differentiation or integration.

Roots and Radicals

Roots and radicals are mathematical concepts that deal with taking a different form of roots, such as square roots or fourth roots, which are covered in this exercise.
The root of a number is essentially its inverse exponent.

• The fourth root of \(x\) is written as \((x^{\frac{1}{4}})\).
• In this exercise, \(\sqrt[4]{\frac{a^3}{81}}\) is the fourth root of \(\frac{a^3}{81}\).

Understanding roots is crucial because it allows you to convert radicals into exponents and vice versa. This conversion is vital for simplifying and solving expressions, particularly those involving quotient rules or powers. By expressing roots as exponents, complex problems become more accessible to solve.