Problem 66 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{x^{8} y^{12}} $$

Step-by-Step Solution

Answer

The simplified form is \( x^2 y^3 \).

Step 1: Understand the Problem

We need to simplify the fourth root of a radical expression \( \sqrt[4]{x^8 y^{12}} \). This means breaking down the expression into a simpler form where the power is reduced, if possible.

Step 2: Apply the Fourth Root to Each Term

The fourth root of a product can be broken down as the product of the fourth roots: \( \sqrt[4]{x^8 y^{12}} = \sqrt[4]{x^8} \cdot \sqrt[4]{y^{12}} \).

Step 3: Simplify the Fourth Root of \(x^8\)

The fourth root of \(x^8\) is \(x^{8/4}\), because \( \sqrt[n]{a^m} = a^{m/n} \). So, \( \sqrt[4]{x^8} = x^2 \).

Step 4: Simplify the Fourth Root of \(y^{12}\)

The fourth root of \(y^{12}\) is \(y^{12/4}\), using the same principle as before. Hence, \( \sqrt[4]{y^{12}} = y^3 \).

Step 5: Combine the Results

Now, multiply the simplified results together: \( x^2 \cdot y^3 \). Hence, the expression is simplified to \( x^2 y^3 \).

Key Concepts

Fourth Root

When we talk about the fourth root of a number, we mean a number that, when multiplied by itself four times, equals the original number. It's like an extension of the square root, but instead of multiplying twice, you multiply four times. For example, the fourth root of 81 is 3 because

• 3 multiplied by itself four times (3 \times 3 \times 3 \times 3) results in 81.

In our exercise, we start with the expression \( \sqrt[4]{x^8 y^{12}} \). The goal is to simplify this using the concept of the fourth root. When simplifying, you divide the exponent by 4 because you are undoing the operation of raising a number to a power of 4. If the exponent is a multiple of 4, like in \( x^8 \), it divides evenly, making calculations straightforward.

• You divide 8 by 4, ending up with \( x^2 \), and 12 by 4, leading to \( y^3 \).

This division process helps in breaking down complex expressions into manageable terms.

Radical Expressions

Radical expressions involve roots, and they include symbols like the square root \( \sqrt{} \), cube root \( \sqrt[3]{} \), and the fourth root \( \sqrt[4]{} \). Simplifying radical expressions is about restructuring them to their simplest forms.
In the example \( \sqrt[4]{x^8 y^{12}} \), it's a fourth root expression, which indicates parts of this expression are "under the radical." To simplify, we separate the expression into parts that can easily be raised to their roots.

• The expression can be split as \( \sqrt[4]{x^8} \cdot \sqrt[4]{y^{12}} \).

Once broken down, each component is treated individually, simplifying them step-by-step. When each part of the radical expression is simplified separately, you multiply them back together to get the simplified version of the original expression. This reduces complexity and confusion, leading to a cleaner and more concise expression.

Exponent Rules

Exponent rules are essential for simplifying expressions with powers. They guide how to handle terms involving exponents, especially when combining, multiplying, or taking roots. Initially, you must understand expressions like \( a^{m/n} \) which represent radical or fractional exponents.
In our task, we use when you take the fourth root of a power like \( \sqrt[4]{x^8} \), it becomes \( x^{8/4} \). This rule shows how dividing an exponent by the root's number transforms it:

• \( a^{m/n} = \sqrt[n]{a^m} \) simplifies as \( a^{m/n} = a^{m/n} \).
• For \( a^0 \), it simplifies to 1, as anything raised to the power of zero is 1.

These rules bring clarity and predictability, making complex operations involving exponents more manageable. Following these exponent laws enables us to simplify radical expressions stepwise, resulting in forms like \( x^2 \) from \( \sqrt[4]{x^8} \) and \( y^3 \) from \( \sqrt[4]{y^{12}} \). Understanding and applying these exponent rules ensures accurate and efficient simplification of any algebraic expression with roots and powers.