Problem 85 - Algebra for College Students

Question

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{3}}{\sqrt[4]{3}}\)

Step-by-Step Solution

Answer

\( 3^{1/12} \)

Step 1: Identify the Exponents

In this problem, we have two radicals: \( \sqrt[3]{3} \) and \( \sqrt[4]{3} \). Let's express them with exponents. Recall that \( \sqrt[n]{a} = a^{1/n} \). Therefore, \( \sqrt[3]{3} = 3^{1/3} \) and \( \sqrt[4]{3} = 3^{1/4} \).

Step 2: Apply the Quotient Rule

Use the quotient rule for exponents, which states that \( \frac{a^{m}}{a^{n}} = a^{m-n} \). In this case, we have the same base, which is 3, in the numerator and denominator. Thus, \( \frac{3^{1/3}}{3^{1/4}} = 3^{1/3 - 1/4} \).

Step 3: Subtract the Exponents

Subtract the exponents: \( \frac{1}{3} - \frac{1}{4} \). To subtract these fractions, find a common denominator. The least common denominator of 3 and 4 is 12. Thus, \( \frac{1}{3} = \frac{4}{12} \) and \( \frac{1}{4} = \frac{3}{12} \). Subtract these: \( \frac{4}{12} - \frac{3}{12} = \frac{1}{12} \).

Step 4: Simplify to Final Expression

Now, we express the result from the subtraction of the exponents, so we have \( 3^{1/12} \). This is already in the simplest radical form since there is no perfect power of 3 that reduces further inside the radical.

Key Concepts

Exponent Rules

Exponent rules are essential tools in algebra that help us manipulate expressions involving powers. One of the most important rules is the quotient rule. This rule states that when you divide like bases, such as dividing \( a^m \) by \( a^n \), you subtract the exponents: \( a^{m-n} \). So, if you have an expression like \( \frac{3^{1/3}}{3^{1/4}} \), you can use the quotient rule to simplify it by subtracting the exponents \( \frac{1}{3} - \frac{1}{4} \). Another useful rule is the power of a power rule, which tells us \( (a^m)^n = a^{m \times n} \). Exponent rules also help when multiplying like bases, where we simply add the exponents: \( a^m \times a^n = a^{m+n} \). Understanding these rules can greatly simplify working with exponential expressions.

Simplifying Radicals

Simplifying radicals is a process that makes expressions involving roots easier to work with. A radical expression involves roots, such as square roots \( \sqrt{} \), cube roots \( \sqrt[3]{} \), or any higher-order roots. To simplify a radical expression, you need to find the prime factorization inside the radical and pull out any perfect powers. However, there are times, like with \( 3^{1/12} \), when the number inside the radical cannot be simplified further.

• Check for perfect powers: Look for factors that are perfect squares in square roots or perfect cubes in cube roots.
  If you find one, pull it out to simplify the expression.
• Combine like radicals: Only combine radicals if they have the same index and the same radicand.

It's helpful to express numbers under radicals with exponents to easily see which can be simplified. Understanding when and how to simplify radicals is key in making mathematical expressions more manageable.

Fractional Exponents

Fractional exponents offer another way to represent roots, like square roots and cube roots, using exponent notation. Instead of writing \( \sqrt[3]{a} \), you can write \( a^{1/3} \), where the numerator is the power and the denominator is the root. This allows us to use exponent rules to simplify calculations.

• Conversion: Convert from radical form to exponential form using \( \sqrt[n]{a} = a^{1/n} \).
• Simplifying: Apply exponent rules to make calculations easier, such as using the quotient rule for dividing like bases.
• Multiple forms: Recognize that fractional exponents provide flexibility, allowing you to switch between different forms of understanding and computation.

For example, using fractional exponents allows the expression \( \frac{\sqrt[3]{3}}{\sqrt[4]{3}} \) to be simplified as \( 3^{1/12} \). This method not only streamlines calculations but also provides a clearer understanding of the relationship between powers and roots.