Problem 118 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that all variables represent positive numbers. $$\sqrt[4]{48 z^{5}}+\sqrt[4]{768 z^{5}}$$

Step-by-Step Solution

Answer

The simplified expression is \(10z \sqrt[4]{3z}\).

Step 1: Simplify Individual Radicals

First, simplify each fourth root separately. For \( \sqrt[4]{48 z^{5}} \), factor 48 as \( 2^4 \times 3 \), which gives us \( \sqrt[4]{2^4 \times 3 \times z^4 \times z} \). Similarly, for \( \sqrt[4]{768 z^{5}} \), factor 768 as \( 2^8 \times 3 \), which gives us \( \sqrt[4]{2^8 \times 3 \times z^4 \times z} \).

Step 2: Take Out Perfect Fourth Powers

Extract the fourth roots of the perfect fourth powers from each radical. For \( \sqrt[4]{2^4 \times 3 \times z^4 \times z} \), take out \( 2z \) (both \( 2^4 \) and \( z^4 \) under the fourth root become 2 and z respectively), leaving \( 2z \sqrt[4]{3z} \). For \( \sqrt[4]{2^8 \times 3 \times z^4 \times z} \), take out \( 8z \), leaving \( 8z \sqrt[4]{3z} \).

Step 3: Combine Like Terms

Notice that both expressions have a common factor of \( z \sqrt[4]{3z} \). Combine the expressions: \( 2z \sqrt[4]{3z} + 8z \sqrt[4]{3z} \), which can be factored as \( (2 + 8)z \sqrt[4]{3z} \).

Step 4: Simplify the Result

Simplify the expression by adding the coefficients: \( (2 + 8) = 10 \). Thus, the expression becomes \( 10z \sqrt[4]{3z} \).

Key Concepts

Simplifying Radicals

Radicals can often seem confusing at first, but once you understand the process of simplifying them, it becomes much easier. In math, a radical is another way of expressing roots. Simplifying a radical means rewriting it in its simplest form, which can make calculations easier.

When you're simplifying radicals, here are a few steps to follow:

• Identify Perfect Powers: The goal is to rewrite the number under the radical as a product of perfect powers and other factors.
• Take Out Perfect Powers: Once you've identified the perfect powers that fit the index of the radical (in our case, the fourth root), extract them out. For example, if you are working with the square root of 16, since 16 is a perfect square (4), you can take it out of the radical.
• Simplify: Multiply all parts outside the radical symbol. Whatever remains inside the radical is already in its simplest form.

By following these steps, any radical expression can be simplified, and calculations can become much easier to handle.

Fourth Root Simplification

The fourth root of a number is all about finding the number that, when multiplied by itself four times, gives the original number. Think of it as a more complex form of taking a square root. The symbol for a fourth root is \(\sqrt[4]{} \). Simplifying a fourth root involves similar steps to simplifying radicals generally but requires one additional detail.

To simplify expressions like \( \sqrt[4]{48z^5} \), follow these steps:

• Factor the Expression: First, break down any numbers into prime factors. For example, 48 can be broken down into \(2^4 \times 3\) and it accompanies \(z^5\) which can be rewritten as \(z^4 \times z\). Thus, \( \sqrt[4]{48z^5} \) simplifies to \( \sqrt[4]{2^4 \times 3 \times z^4 \times z} \).
• Extract Perfect Fourth Powers: If any of the components can be written as perfect fourth powers, that is an opportunity to extract them out of the root. Here, both \(2^4\) and \(z^4\) are perfect fourth powers, so they come out as 2 and \(z\) respectively.
• Recombine: What remains under the root can no longer be simplified, in this case \( \sqrt[4]{3z} \), and you multiply outside components like 2 and \(z\) back into a single product: \(2z \sqrt[4]{3z} \).

Fourth root simplification can streamline expressions and is particularly useful in algebra and higher mathematics.

Factoring

Factoring is a powerful tool when it comes to simplifying algebraic expressions or finding solutions to equations. It involves rewriting a number or expression as a product of factors, which are smaller or simpler components. In the context of simplifying radicals and algebraic expressions, factoring is essential.

Here's how to factor expressions effectively:

• Find Greatest Common Factor (GCF): Identify the largest number or expression that can be divided out of each term in the expression. This is often the first step in simplifying expressions.
• Factor within Radicals: For expressions like \( \sqrt[4]{768z^5} \), break down the numbers into their prime factors, such as \( 2^8 \times 3\) in the example. This makes it easy to identify and extract perfect powers, which simplifies your root.
• Refactor and Simplify: Once you've taken factors out, rewrite the expression in its simplest form. This often involves combining like terms, as we did when recognizing both expressions \( 2z \sqrt[4]{3z} + 8z \sqrt[4]{3z} \) share the common factor \( z \sqrt[4]{3z} \), allowing us to factor further to \((2 + 8)z \sqrt[4]{3z}\).

Factoring is crucial because it reveals the structure of expressions, helping you simplify and solve them more easily. It’s one of the basic, yet most transformative, tools in algebra.