Problem 24 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt[4]{27} \cdot \sqrt[4]{3} $$

Step-by-Step Solution

Answer

The product simplifies to 3.

Step 1: Multiply the Radicands

The expression is \( \sqrt[4]{27} \cdot \sqrt[4]{3} \). When multiplying two radicals with the same index, you can multiply the radicands together. So, \( \sqrt[4]{27} \cdot \sqrt[4]{3} = \sqrt[4]{27 \times 3} \).

Step 2: Simplify the Multiplication inside the Radicand

Multiply the numbers inside the radicand: \( 27 \times 3 = 81 \). So, we rewrite the expression as \( \sqrt[4]{81} \).

Step 3: Evaluate the Fourth Root

The expression is now \( \sqrt[4]{81} \). Since \( 81 = 3^4 \), the fourth root of 81 is 3, because \( (3)^4 = 81 \). Therefore, \( \sqrt[4]{81} = 3 \).

Key Concepts

Radicals

Radicals are a way to express roots of numbers. A radical is represented with the symbol \( \sqrt{} \), which is known as the "radical sign". The number inside the radical sign is called the "radicand". The index, written as a small number to the left of the radical sign (such as 4 in \( \sqrt[4]{} \)), indicates which root you are taking. For example, \( \sqrt[4]{16} \) is asking for the fourth root of 16.

• Radicand: the number inside the radical sign.
• Index: indicates the degree of the root.
• Common radical forms include square roots (index of 2), cube roots (index of 3), and fourth roots (index of 4).

Understanding how radicals work helps in simplifying complex expressions and solving equations involving roots.

Multiplication of Radicals

When you multiply two radicals with the same index, you can combine the radicands under a single radical. For example, consider \( \sqrt[4]{a} \cdot \sqrt[4]{b} \). To multiply these, you can rewrite them as \( \sqrt[4]{a \times b} \). It's important to ensure both radicals have the same index for this operation to work.

• This method simplifies the process by reducing the number of radicals in an expression.
• It helps to deal with complex expressions more easily, especially when simplifying large calculations.
• Always make sure the radicands remain positive to keep the expression real.

This is an efficient technique for solving algebraic problems with radicals.

Indices

Indices refer to the power to which a number is raised, or, in the case of radicals, the root that you're taking. In the expression \( \sqrt[4]{81} \), the 4 is the index, indicating a fourth root. Indices play a crucial role in both multiplication and simplification of radicals.

• The index tells you what degree of root you are calculating.
• In simplifying radicals, it's often necessary to convert between indices and exponential forms (e.g., \( 81 = 3^4 \)), as it allows you to directly determine roots.
• This conversion is key for simplifying radicals and making calculations approachable and easier to compute.

Understanding indices is central to mastering operations involving radicals.

Simplifying Expressions

Simplifying expressions involves breaking down complex expressions into their simplest form. This process often involves working with radicals, indices, and other algebraic operations.
To simplify a radical expression:

• Combine like radicals, if possible.
• Simplify the radicand by factoring into perfect powers that match the index.
• Apply the operation of finding roots based on indices.
  For instance, in \( \sqrt[4]{81} \), you recognize \( 81 = 3^4 \) and thus \( \sqrt[4]{81} = 3 \).

Simplifying expressions helps reveal the core components of a mathematical problem, making it easier to solve and understand. It's a vital skill in mathematics that aids in tackling more advanced topics.