Problem 12 - Intermediate Algebra

Question

Use the product rule to multiply. See Example \(I\). $$ \sqrt[4]{a b^{2}} \cdot \sqrt[4]{27 a b} $$

Step-by-Step Solution

Answer

The simplified expression is \(\sqrt[4]{27a^2b^3}\).

Step 1: Express in Radical Form

Rewrite both expressions using exponents to better handle them. The original terms are: \(\sqrt[4]{a b^{2}} \quad \text{and} \quad \sqrt[4]{27 a b}\)This becomes: \((a b^{2})^{1/4} \quad \text{and} \quad (27 a b)^{1/4}\)

Step 2: Use the Product Rule for Radicals

The product rule for radicals states that \(\sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{xy}\). Apply this to combine the radicals:\[((a b^{2}) \cdot (27 a b))^{1/4}\]

Step 3: Simplify Inside the Radical

Simplify the expression inside the radical. Multiply the expressions:\(a \, b^{2} \cdot 27 \, a \, b = 27 \, a^{2} \, b^{3}\)

Step 4: Write the Final Expression with a Single Radical

Substitute the simplified expression back under a single radical:\[\sqrt[4]{27 a^{2} b^{3}}\]

Step 5: Simplify If Possible

Attempt to simplify the radical further if possible. Each term in the radical should be checked for fourth powers or factorizations that allow simplification.Currently, without additional information on factorization possibilities of 27 and whether a, b have roots, \(\sqrt[4]{27 a^{2} b^{3}}\) is the simplest form.

Key Concepts

Radical Form

Understanding radical form is essential when working with expressions involving roots, such as square, cube, or fourth roots. A radical expression consists of a radicand, which is the number under the radical sign, and the index, which indicates the degree of the root. For example, in the expression \(\sqrt[4]{a b^{2}}\), the index is 4, indicating a fourth root, and \(a b^{2}\) is the radicand. Radicals can be converted to exponential form, which makes it easier to handle algebraic manipulations. In exponential form, the radical \(\sqrt[n]{x}\) can be expressed as \(x^{1/n}\). Thus, \(\sqrt[4]{a b^{2}}\) becomes \((a b^{2})^{1/4}\).Converting between these forms helps in simplifying complex radical expressions and applying rules like the product rule effectively.

Simplifying Radicals

Simplifying radicals is the process of making a radical expression as simple as possible, which often involves removing any perfect powers within the radicand. The given problem deals with radicals, \(\sqrt[4]{a b^{2}}\) and \(\sqrt[4]{27 a b}\), which are initially simplified by multiplying under one radical using the product rule. The steps to simplify the problem include:

• Combining the radicands under a single radical: \((a b^{2}) \cdot (27 a b)\), resulting in \(27 a^{2} b^{3}\).
• Checking for any perfect fourth powers inside: In this case, there aren't straightforward fourth powers for 27 or the other terms, hence the expression remains in its combined form \(\sqrt[4]{27 a^{2} b^{3}}\).

Attempting further simplification involves checking if any factors like \(a\) or \(b\) can lead to simpler expressions if they have perfect fourth powers. However, with the provided expression, it remains as is.

Exponents

Exponents play a crucial role when working with radicals, as they provide a convenient way to express roots and manipulate the expressions algebraically. In the context of the exercise, radicals were converted into exponents for simplification. The expressions \(\sqrt[4]{a b^{2}}\) and \(\sqrt[4]{27 a b}\) are rewritten using exponents as \((a b^{2})^{1/4}\) and \((27 a b)^{1/4}\) respectively. This transformation makes it easier to apply the product rule for radicals, allowing these expressions to be combined into \(((a b^{2}) \cdot (27 a b))^{1/4}\).Understanding the basic properties of exponents, such as \((x^{m})^{n} = x^{m \cdot n}\) and how to multiply or divide expressions with the same base, simplifies the handling of expressions. In the given problem, this understanding is crucial for moving smoothly between radical and exponential forms to arrive at the most simplified expression.