Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The product rule for radicals: \(\sqrt[n]{a b}=\quad\) In words, the nth root of the ____ of two numbers is equal to the product of their \(nth\) ____ .

Step-by-Step Solution

Answer

The nth root of the product of two numbers is equal to the product of their nth roots.

Step 1: Understand the Product Rule for Radicals

The product rule for radicals states that the nth root of a product of two numbers can be expressed as the product of the nth root of each number. Specifically, \( \sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b} \). This rule allows us to simplify expressions involving roots by breaking them apart into the roots of each factor in the product.

Step 2: Identify the Missing Words

Given the statement "The nth root of the ____ of two numbers is equal to the product of their nth ____", we need to fill in the blanks by relating this to the product rule. The first blank should describe what is inside the root, which is a product. The second blank should describe what we take of each number separately, which is the root.

Step 3: Fill in the Blanks with the Correct Terms

Using the product rule for radicals \( \sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b} \), we conclude that: "The nth root of the **product** of two numbers is equal to the product of their nth **roots**." Therefore, the missing words are 'product' for the first blank and 'roots' for the second blank.

Key Concepts

Product Rule for Radicals

The product rule for radicals is a fundamental concept in algebra that helps in simplifying expressions involving roots. This rule provides a straightforward way to break down the nth root of a multipled expression into simpler components. Essentially, the rule states that the nth root of the product of two numbers is the product of the nth roots of each number. This can be mathematically expressed as:

• \[ \sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b} \]

By using this rule, we can simplify complex radical expressions. For example, if you are given the expression \( \sqrt{28} \), breaking it into factors like 4 and 7 allows you to express it as \( \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2 \sqrt{7} \). This technique can make calculations easier and more intuitive.

Nth Root

The concept of the nth root extends the idea of square roots to other roots, such as cube roots and fourth roots. The nth root of a number is a value that, when raised to the power of n, yields the original number. Mathematically, for a number \( x \), the nth root is represented as \( \sqrt[n]{x} \).

• This means: \( \sqrt[n]{x} = y \) implies \( y^n = x \).
• For example, \( \sqrt[3]{27} = 3 \) because \( 3^3 = 27 \).

Understanding nth roots is crucial for solving radical expressions and forms the basis of applying the product rule for radicals. Whether it's the square root or any other root, knowing how to work with nth roots will guide you in simplifying more complicated radical expressions.

Simplifying Radical Expressions

Simplifying radical expressions involves breaking down roots into their simplest form. This often requires using the product rule for radicals or recognizing when and how to simplify the factors under the root.
Let's consider the process:

• First, identify any perfect squares or cubes within the radical.
• Apply the product rule to separate these factors. For example, for \( \sqrt{20} \), rewrite it as \( \sqrt{4 \times 5} \) which becomes \( \sqrt{4} \times \sqrt{5} = 2 \sqrt{5} \).

This simplification is essential to solve problems more efficiently. It's not only about reducing a number under a radical sign but also transforming the expression into an easily manageable form. This skill is particularly valuable when handling complex algebraic expressions, as it leads to cleaner and more precise results.