Problem 86 - Algebra for College Students

Question

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt{2}}{\sqrt[3]{2}}\)

Step-by-Step Solution

Answer

The answer is \( \sqrt[6]{2} \).

Step 1: Write the Fraction with Radical Notation

The given expression is \( \frac{\sqrt{2}}{\sqrt[3]{2}} \). Here, the numerator is a square root and the denominator is a cube root of the same base, 2.

Step 2: Consolidate the Radical Expression using Exponents

Convert each radical into an exponent. Recall that \( \sqrt{a} = a^{1/2} \) and \( \sqrt[3]{a} = a^{1/3} \). Therefore, the expression becomes \( \frac{2^{1/2}}{2^{1/3}} \).

Step 3: Simplify the Expression Using Properties of Exponents

Use the property \( \frac{a^m}{a^n} = a^{m-n} \) to simplify. Subtract the exponents: \( 2^{1/2 - 1/3} = 2^{3/6 - 2/6} = 2^{1/6} \).

Step 4: Express the Result in Simplest Radical Form

Since \( 2^{1/6} \) is the sixth root of 2, you can write this as \( \sqrt[6]{2} \) as the simplest radical form.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. If you have a number raised to an exponent, it simply means you multiply the number, known as the base, by itself a certain number of times, indicated by the exponent. For example, in the case of \( a^n \), \( a \) is multiplied by itself \( n \) times.

Exponents can be positive, negative, or even fractions. Understanding fractional exponents is key when dealing with radical expressions. A fractional exponent, such as \( a^{1/2} \), corresponds to taking the square root of \( a \), while \( a^{1/3} \) corresponds to taking the cube root. These represent roots in exponent form and are crucial in simplifying radical expressions. By converting radicals to exponents, you can easily apply exponent rules to simplify expressions, which helps in finding their simplest form.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. These are represented with the radical symbol (√). For example, \( \sqrt{2} \) is the square root of 2, and \( \sqrt[3]{2} \) is the cube root.

Understanding radical expressions allows you to manipulate and simplify them effectively. Converting radicals to exponents, as seen in the exercise, aids in simplifying because exponent rules are easier to apply than those for radicals.

• Square roots correspond to an exponent of \( \frac{1}{2} \)
• Cube roots correspond to an exponent of \( \frac{1}{3} \)

Using these conversions, radical expressions can be rewritten as exponents, which is helpful for calculations and simplifications.

Properties of Exponents

Properties of exponents are useful tools for simplifying expressions, including radical expressions. A crucial property used in this context is \( \frac{a^m}{a^n} = a^{m-n} \). This property allows you to divide like bases by subtracting the exponents.

For example, in the original exercise, using \( \frac{2^{1/2}}{2^{1/3}} \) becomes \( 2^{1/2-1/3} \), simplifying it to \( 2^{1/6} \).

• To subtract fractional exponents, find a common denominator.
• The property \( a^m \times a^n = a^{m+n} \) can be used if you're multiplying bases.
• Mastering these properties permits efficient handling and simplification of radical and exponential expressions.

By using these properties, radicals can often be simplified, leading to expressions that are easier to work with and solve.