Problem 38 - Algebra and Trigonometry

Question

\(29-38=\) Simplify the expression. Assume that the letters denote any real numbers. $$ \sqrt[4]{x^{4} y^{2} z^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(x \sqrt{y} \sqrt{z}\).

Step 1: Understand the Fourth Root

The expression \( \sqrt[4]{x^4 y^2 z^2} \) represents the fourth root of the product \(x^4 y^2 z^2\). The fourth root of a variable to the fourth power, such as \( x^4 \), simplifies directly to the variable itself because \((x^4)^{1/4} = x\).

Step 2: Simplify Each Component

Break down the expression under the fourth root: \((x^4 y^2 z^2)^{1/4}\). Apply the rule \((a^m)^{n} = a^{mn}\) to each component: - \((x^4)^{1/4} = x^{4/4} = x\)- \((y^2)^{1/4} = y^{2/4} = y^{1/2} = \sqrt{y}\)- \((z^2)^{1/4} = z^{2/4} = z^{1/2} = \sqrt{z}\).

Step 3: Combine the Simplified Components

Combine the simplified results: - The fourth root of \(x^4\) is \(x\).- The fourth root of \(y^2\) is \(\sqrt{y}\).- The fourth root of \(z^2\) is \(\sqrt{z}\). Thus, the expression simplifies to: \(x \sqrt{y} \sqrt{z}\).

Key Concepts

Fourth Root Simplification

When dealing with the fourth root simplification, we're referring to the process of finding a number that, when multiplied by itself four times, yields the original number under the radical. The expression \( \sqrt[4]{x^4 y^2 z^2} \) involves simplifying the product of variables raised to different powers under a fourth root.

Here's how you can approach simplifying such expressions:

• If the exponent of a variable inside the radical is exactly \(4\), it simplifies directly to that variable. For instance, \((x^4)^{1/4} = x\) because taking the fourth root of \(x^4\) cancels the exponent of 4.
• For variables with even but not fourth-power exponents like \(y^2\), dividing the exponent by 4 results in a fractional exponent. Thus, \((y^2)^{1/4} = y^{1/2}\), which can also be written as \(\sqrt{y}\).
• This rule applies similarly to \(z^2\), leading to \(\sqrt[4]{z^2} = \sqrt{z}\).

Simplifying the fourth root requires understanding both integer and fractional exponents. Identifying components that simplify directly from the fourth root is often your first step.

Exponent Rules

Exponent rules are crucial to simplifying expressions involving powers. They provide a framework to manipulate and simplify expressions like \( (x^4 y^2 z^2)^{1/4} \). These rules include:

• Power of a Power Rule: States that \((a^m)^n = a^{m \times n}\). This allows you to multiply the exponents when raising a power to another power. For example, \((y^2)^{1/4}\) becomes \(y^{2/4} = y^{1/2}\).
• Product of Powers Rule: Indicates that \(a^m \cdot a^n = a^{m+n}\). This rule isn't needed directly in our fourth root case, but it's helpful when combining terms of the same base.
• Fractional Exponents: Reflect roots, where \(a^{1/n} = \sqrt[n]{a}\). A fractional exponent like \(1/4\) is equivalent to a fourth root, and this is essential when interpreting an expression like \(z^{1/2}\) as \(\sqrt{z}\).

Practicing these rules will help you tackle more complex expressions effortlessly.

Radical Expressions

Radical expressions involve roots, represented with the radical symbol. The expression \(\sqrt[4]{x^4 y^2 z^2}\) is a type of radical expression, specifically a fourth root. To simplify radical expressions:

• Identify the root type and realize that it's related to an exponent fraction (e.g., fourth root corresponds to a \(1/4\) exponent).
• Break down each factor of the expression into its powers as we did with \(x^4, y^2,\) and \(z^2\).
• Apply the relevant exponent rules to simplify each term individually, then combine them for the final expression.

Radical expressions can often be intimidating, but with practice, breaking them into simpler parts becomes second nature. The key is understanding how to connect the radical form with its corresponding fractional exponent form, leading to consistent simplification.