Problem 4 - Algebra A Combined Function

Question

Use the product rule to multiply. Assume that all variables represent positive real numbers. $$ \sqrt[4]{27} \cdot \sqrt[4]{3} $$

Step-by-Step Solution

Answer

The product is 3.

Step 1: Understand the Expression

The given expression is \( \sqrt[4]{27} \cdot \sqrt[4]{3} \). We will use the product rule for radicals to simplify this expression, which states that \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \).

Step 2: Apply the Product Rule

Using the product rule, we combine the terms under a single radical: \( \sqrt[4]{27} \cdot \sqrt[4]{3} = \sqrt[4]{27 \times 3} \). Simplifying inside the radical gives us \( \sqrt[4]{81} \).

Step 3: Simplify the Radical Expression

We need to simplify \( \sqrt[4]{81} \). To do this, express 81 as a power of its prime factors: \( 81 = 3^4 \). Thus, \( \sqrt[4]{81} = \sqrt[4]{3^4} \).

Step 4: Evaluate the Radical

When \( n \) in \( \sqrt[n]{a^n} \) matches the exponent on \( a \), the expression simplifies to \( a \). Therefore, \( \sqrt[4]{3^4} = 3 \).

Key Concepts

Simplifying Radicals

To simplify radicals, we need to find ways to express a given radical in its simplest form. A radical is an expression that includes a root, like square roots or fourth roots. Simplifying them involves reducing the expression so it's easier to understand and work with.

• First, understand the type of root you are working with. For example, the square root, cube root, or fourth root.
• Identify and break down the number inside the radical to its prime factors. This helps in simplifying.
• Apply the rules of radicals. Use the product rule or quotient rule, depending on whether you are multiplying or dividing radicals.

In our specific example, we had to simplify \( \sqrt[4]{81} \). By expressing 81 as \( 3^4 \), it became evident that simplifying this radical involved acknowledging that the fourth root and the exponent cancel each other, resulting in 3.
Practicing simplifying radicals will make these steps more intuitive and make solving problems quicker.

Exponents and Roots

Exponents and roots are two sides of the same coin. Understanding their relationship is crucial in simplifying expressions with radicals.

• Exponents indicate how many times a number, known as the base, is to be multiplied by itself. For example, \( 3^4 \) represents \( 3 \times 3 \times 3 \times 3 \).
• Roots, on the other hand, are the inverse of exponents. The fourth root \( \sqrt[4]{a} \) asks what number, when raised to the power of 4, will result in \( a \).
• A core concept is that \( (a^n)^{1/n} = a \). This shows the balancing act between exponents and roots.

Applying this to our given exercise, simplification involved recognizing that \( \sqrt[4]{3^4} \) simplifies directly to 3. Knowing how exponents correspond with roots aids in this simplification process.

Radical Expressions

Radical expressions can look complex, but they become much easier to handle once you understand the underlying principles. Let's break this down:

• Radical expressions involve roots represented by the radical symbol (√) and can include any level of root, not just square roots.
• The product rule for radicals is a handy tool. It states that \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \). This allows you to simplify a product of radicals into a single radical.
• Keep in mind, this rule only applies when the radicals have the same index, meaning they are of the same degree.

In the original exercise, utilizing the product rule allowed us to combine \( \sqrt[4]{27} \) and \( \sqrt[4]{3} \) into a single radical, \( \sqrt[4]{81} \), which was then simplified. Understanding these expressions makes solving and simplifying much easier.