Problem 81 - Algebra A Combined Function

Question

$$ -\sqrt[3]{\frac{64 a^{3}}{b^{9}}} $$$$ \sqrt[4]{\frac{x^{4}}{16}} $$

Step-by-Step Solution

Answer

Cubic root: \(-\frac{4a}{b^3}\), fourth root: \(\frac{x}{2}\).

Step 1: Simplify the Radicand of the Cubic Root

Inside the cubic root, identify the components that can be simplified. The expression is \(-\sqrt[3]{\frac{64 a^{3}}{b^{9}}}\). Rewrite \(64\) as \(4^3\) and \(b^9\) as \((b^3)^3\), so you have \(-\sqrt[3]{\frac{4^3 a^3}{(b^3)^3}}\).

Step 2: Simplify the Cubic Root

Take the cubic root of the components individually. The cube root of \(4^3\) is 4, and the cube root of \(a^3\) is \(a\), and \(b^9\) is \((b^3)^3\), whose cube root is \(b^3\). Therefore, the expression simplifies to \(-\frac{4a}{b^3}\).

Step 3: Simplify the Radicand of the Fourth Root

In the expression \( \sqrt[4]{\frac{x^{4}}{16}} \), notice that \(16\) can be rewritten as \(2^4\). The expression becomes \(\sqrt[4]{\frac{x^4}{2^4}}\).

Step 4: Simplify the Fourth Root

Take the fourth root of the components. The fourth root of \(x^4\) is \(x\), and the fourth root of \(2^4\) is 2. So the expression simplifies to \(\frac{x}{2}\).

Step 5: Combine Simplified Results

We've solved each expression independently. The final results are \(-\frac{4a}{b^3}\) for the cubic root and \(\frac{x}{2}\) for the fourth root.

Key Concepts

Radicand

The term **radicand** refers to the number or expression inside a radical symbol. When dealing with cube roots or fourth roots, identifying the radicand is crucial to simplifying the expression. In our example, for the cube root expression

• We have \(-\sqrt[3]{\frac{64 a^{3}}{b^{9}}}\), where \(\frac{64 a^{3}}{b^{9}}\) is the radicand.

The key to simplifying this lies in breaking down the radicand into powers that are easier to manage. Similarly, for the fourth root expression, the radicand is \(\frac{x^{4}}{16}\).

• To simplify it, think about how you can express its components in a factorable way, like writing 16 as \(2^4\).

Understanding the radicand enables further simplification, leading us directly to the next step: calculating the root.

Fourth Root

If you've ever wondered what a **fourth root** is, think of it as the value that, when multiplied by itself four times, gives you the original number. In mathematical expressions, this is typically represented using radical symbols.
In the given example of \(\sqrt[4]{\frac{x^{4}}{16}}\), the objective here is to simplify this expression into a form that's easy to interpret.
To solve this, we can rewrite 16 as \(2^4\), making it easy to factor out powers and find the fourth root individually. The goal is to identify components like \(x^4\) and \(2^4\) which give whole number roots, making simplification straightforward.

• The fourth root of \(x^4\) simplifies to x.
• The fourth root of \(2^4\) simplifies to 2.

Combining these results, the expression simplifies to \(\frac{x}{2}\). Being comfortable with finding fourth roots will significantly aid in handling more complex algebraic expressions.

Exponentiation

**Exponentiation** is a fundamental mathematical operation that involves raising numbers, known as the bases, to certain powers, or exponents. In the context of our problem, exponentiation plays a crucial role in simplifying expressions.
For instance, when simplifying \(\frac{64 a^{3}}{b^{9}}\), understanding that 64 can be written as \(4^3\) is key. This not only simplifies the expression but also makes finding cube roots manageable.

• A critical step is recognizing everything in terms of powers—the fact that \(b^9\) simplifies to \( (b^3)^3 \) aids future simplification.

When you take the **exponentiation** apart in the context of roots, it allows complex expressions to be broken down, revealing the underlying simplicity. Knowing how to rewrite these terms can make complicated algebra feel much more approachable.

Algebraic Expressions

The term **algebraic expressions** refers to mathematical phrases that include numbers, variables, and arithmetic operations. In our exercise, both expressions involve algebraic components that require understanding and manipulation.
Consider the cube root expression \(-\sqrt[3]{\frac{64 a^{3}}{b^{9}}}\). This expression combines constants and variables in a fraction—a common format in algebra.

• By simplifying the constituents of the expression, it becomes \(-\frac{4a}{b^3}\), which is far easier to interpret and work with.

The fourth root expression, \(\sqrt[4]{\frac{x^{4}}{16}}\), similarly requires understanding how to break down and recombine algebraic terms for simplification.

• This results in a simpler expression, \(\frac{x}{2}\).

Mastering simplification of algebraic expressions involves not only knowing rules but also knowing when and how to apply them. This makes navigating through algebra in any form a more navigable journey.