Problem 64 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{10}}{\sqrt[3]{2}}\)

Step-by-Step Solution

Answer

\( \sqrt[3]{5} \)

Step 1: Identify the Given Expression

The given expression is \( \frac{\sqrt[3]{10}}{\sqrt[3]{2}} \). This expression represents the division of two cube roots.

Step 2: Apply the Quotient Rule for Roots

According to the quotient rule for roots, \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \). Applying the rule here gives us: \( \sqrt[3]{\frac{10}{2}} \).

Step 3: Simplify the Fraction Inside the Radicand

Simplify the fraction inside the cube root: \( \frac{10}{2} = 5 \). Now the expression becomes: \( \sqrt[3]{5} \).

Step 4: Write the Simplified Expression

After simplifying, the division of the cube roots results in \( \sqrt[3]{5} \). No further simplification is possible since 5 is a prime number.

Key Concepts

Cube Roots

In mathematics, a cube root is a number that, when multiplied by itself three times, gives you the original number. For example, if you have a number written as \( \sqrt[3]{x} \), it means you need a number that, when raised to the power of three, equals \( x \). This is different from square roots, which involve two multiplications. Cube roots are often used when dealing with volume or when simplifying expressions involving exponents. Understanding cube roots can make handling expressions like \( \sqrt[3]{10} \) easier, as you are essentially looking for a number that makes the equation hold true when cubed. Cube roots follow the rule that \( \sqrt[3]{x} = x^{1/3} \), allowing conversion between radical and exponential forms of expressions. Recognizing this relationship is crucial when simplifying expressions or working with fractions that involve cube roots.

Division of Radicals

Dividing radicals involves certain rules and properties that can make the process straightforward. The key to dividing radicals, such as cube roots, is using the Quotient Rule for Radicals. This rule states:

• If you have \( \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \), it can be rewritten as \( \sqrt[3]{\frac{a}{b}} \).
• This allows you to combine the radicands under a single radical sign and subsequently simplify.

In practical terms, take an expression like \( \frac{\sqrt[3]{10}}{\sqrt[3]{2}} \). According to the rule, you rewrite it as \( \sqrt[3]{\frac{10}{2}} \), which simplifies the operation considerably. This approach helps streamline the process of handling complex fractions and allows for clearer and more efficient calculations. It's a useful approach to know, especially in algebra and calculus, where radicals frequently appear.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form. For cube roots, this means making the expression as simple as possible while ensuring it is still equivalent to the original. Once we apply the quotient rule for division as in \( \sqrt[3]{\frac{10}{2}} \), the next step is simplifying the fraction within the cube root. Performing this division gives us \( \sqrt[3]{5} \). At this point, checking for further simplification is necessary. For cube roots, you look for perfect cubes or prime factors. Since 5 is a prime number and does not have any perfect cube factors, \( \sqrt[3]{5} \) is in its simplest form. Taking these steps helps ensure you don't overcomplicate the expression, making it manageable and easier to understand for anyone reviewing or checking calculations. By practice, simplifying radicals becomes a useful skill in solving various mathematical problems efficiently.