Problem 42 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{x^{8} y^{7} z^{9}}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^2 y z^2 \sqrt[4]{y^3 z}\).

Step 1: Understand the Problem

We need to simplify the expression \(\sqrt[4]{x^{8} y^{7} z^{9}}\). This means we have to find a simpler form of this fourth root expression so that it involves either whole powers or simplest radical expressions.

Step 2: Apply the Rule of Simplification

To simplify the fourth root, use the property \(\sqrt[n]{a^m} = a^{m/n}\). Apply this to each term separately for the expression inside the root: 1. \(x^{8/4}\)2. \(y^{7/4}\)3. \(z^{9/4}\).

Step 3: Simplify Each Term Separately

Compute the simplified form of each term using the exponential rule:1. \(x^{8/4} = x^2\).2. \(y^{7/4} = y^{1 + 3/4} = y \cdot y^{3/4}\). 3. \(z^{9/4} = z^{2 + 1/4} = z^2 \cdot z^{1/4}\).

Step 4: Write the Simplified Expression

Combine these results: \(x^2 \cdot y \cdot y^{3/4} \cdot z^2 \cdot z^{1/4}\).

Step 5: Final Simplified Form

The simplest form is \(x^2 y z^2 \sqrt[4]{y^3 z}\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and in our case, fourth roots. These expressions require understanding how to simplify them while maintaining their mathematical equivalence. To simplify a radical expression, you often aim to rewrite it in a simpler form without altering its value. This might involve finding the smallest possible whole number exponents inside the radical.

For instance, consider the fourth root of an expression like \( \sqrt[4]{x^8 y^7 z^9} \). Here, the radical indicates the fourth root of a product of variables raised to various powers. The goal in simplifying is to express this in a form that's more straightforward, possibly pulling whole powers of the variables out from under the radical sign, leaving the remainder in simplest radical form.

Exponential Rules

Exponential rules are integral when simplifying expressions with radicals. A critical rule to remember is \( \sqrt[n]{a^m} = a^{m/n} \). This rule allows you to convert the radical into an exponential form, making it easier to manipulate mathematically.

Applying this to each component in our original expression \( \sqrt[4]{x^{8} y^{7} z^{9}} \) gives us separate terms:

• For \( x^8 \), the fourth root is \( x^{8/4} = x^2 \).
• For \( y^7 \), it becomes \( y^{7/4} \), which can be expressed as \( y \cdot y^{3/4} \).
• For \( z^9 \), it simplifies to \( z^{9/4} \) or \( z^2 \cdot z^{1/4} \).

These transformations make it easier to see which parts of the expression can be simplified outside the radical and which parts need to remain under it. By applying the exponential rule systematically, each part becomes more manageable.

Algebraic Simplification

Algebraic simplification involves reducing an expression to its most concise form, removing complexity while preserving equality. Start by addressing each component of your expression, ensuring all terms outside of roots are minimized to whole numbers when possible. In the expression from our problem, we utilize both the exponential rule and thoughtful organization of terms.

Combining our simplified terms gives:

• \( x^2 \) comes straightforwardly as a factor outside the radical.
• \( y^{7/4} \) becomes \( y \cdot y^{3/4} \), simplifying the whole power \( y \).
• \( z^{9/4} \) simplifies to \( z^2 \cdot z^{1/4} \), leaving \( z^2 \) outside.

Thus, the expression simplifies neatly into \( x^2 y z^2 \sqrt[4]{y^3 z} \), taking whole powers outside and leaving any remainders as simpler radicals. The goal of simplification is always clarity and the reduction of complexity without changing the expression's value.