Problem 3 - Intermediate Algebra

Question

Use the product rule to multiply. See Example \(I\). \(\sqrt[4]{8} \cdot \sqrt[4]{2}\)

Step-by-Step Solution

Answer

The product \(\sqrt[4]{8} \cdot \sqrt[4]{2}\) simplifies to 2.

Step 1: Express in Exponential Form

First, we need to express each radical in exponential form. The fourth root of a number can be written using an exponent of \(\frac{1}{4}\). Thus, \(\sqrt[4]{8}\) can be written as \(8^{\frac{1}{4}}\) and \(\sqrt[4]{2}\) as \(2^{\frac{1}{4}}\).

Step 2: Apply the Product Property of Exponents

The product rule states that \(a^m \cdot a^n = a^{m+n}\). Applying this rule, we multiply \(8^{\frac{1}{4}}\) and \(2^{\frac{1}{4}}\) to get:\[((8 \cdot 2)^{\frac{1}{4}}) = 16^{\frac{1}{4}}.\]

Step 3: Simplify Result

Now, simplify \(16^{\frac{1}{4}}\). We know that 16 is a power of 2 (specifically \(2^4\)), so:\[16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}}.\]According to the rule \((a^{m})^{n} = a^{m \cdot n}\), this simplifies to:\[2^{4 \cdot \frac{1}{4}} = 2^{1} = 2.\]

Key Concepts

Exponential Form

In the world of mathematics, expressing numbers in exponential form is a handy skill. This technique allows us to simplify complex expressions, especially when working with roots and powers. For instance, the fourth root of a number like 8 can be rewritten as the base number raised to the power of a fraction. Here, since we are dealing with a fourth root, we write it as the number raised to the power of \( \frac{1}{4} \). So, in our example:

• \( \sqrt[4]{8} \) becomes \( 8^{\frac{1}{4}} \)
• \( \sqrt[4]{2} \) becomes \( 2^{\frac{1}{4}} \)

Why is this useful? It allows us to perform operations like multiplication and division more easily, using the powerful rules of exponents. By converting roots into fractional exponents, we unlock a range of mathematical tools that make solving problems much smoother.

Product Property of Exponents

The product property of exponents is a fundamental rule that simplifies the multiplication of exponential terms with the same base. This rule states that when you multiply two exponents with the same base, you simply add the exponents. The general formula is \( a^m \cdot a^n = a^{m+n} \).In our specific example, we apply this rule:

• Multiply \( 8^{\frac{1}{4}} \) and \( 2^{\frac{1}{4}} \)
• Combine them under a single fractional exponent with the sum of the exponents: \((8 \cdot 2)^{\frac{1}{4}} = 16^{\frac{1}{4}}\)

This approach simplifies expressions involving roots and allows us to reduce the number of steps in calculations. By understanding and applying this property, you'll find it much easier to solve exponential equations accurately.

Simplifying Radicals

Simplifying radicals often involves reducing them to a more manageable form using known powers. In our example, once the expression is condensed to \( 16^{\frac{1}{4}} \), the next step is simplification.Here's how:

• Recognize that 16 is a power of 2, specifically \( 2^4 \)
• Rewrite the expression as \((2^4)^{\frac{1}{4}}\)

According to the rule \((a^m)^n = a^{m\cdot n}\), simplify it further:

• \( 2^{4 \cdot \frac{1}{4}} = 2^1 = 2 \)

This simplification shows how radicals connected to known powers can be handled efficiently. Knowing the rules for manipulating exponents and radicals can turn seemingly complex calculations into straightforward tasks.