Problem 82 - Algebra for College Students

Question

For Problems \(81-90\), perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\sqrt{2} \sqrt[4]{2}\)

Step-by-Step Solution

Answer

\( \sqrt[4]{8} \)

Step 1: Convert the Radicals to Exponential Form

First, express each radical using exponents: \( \sqrt{2} \) can be written as \( 2^{1/2} \), and \( \sqrt[4]{2} \) can be written as \( 2^{1/4} \).

Step 2: Multiply Exponents with the Same Base

When multiplying exponents with the same base, you add the exponents: \( 2^{1/2} \times 2^{1/4} = 2^{(1/2 + 1/4)} \).

Step 3: Add the Fractions

To add \( \frac{1}{2} \) and \( \frac{1}{4} \), find a common denominator. The common denominator is 4, so \( \frac{1}{2} \) becomes \( \frac{2}{4} \). Hence, \( \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \).

Step 4: Express as Simplest Radical Form

Convert the result back to radical form: \( 2^{3/4} = \sqrt[4]{2^3} \). Simplifying \( 2^3 \) gives 8, so the answer is \( \sqrt[4]{8} \).

Key Concepts

Exponents

Exponentiation is a mathematical operation that involves raising a number (the base) to a certain power (the exponent). When we express a number in exponential form, we indicate how many times the base is multiplied by itself. For example, in the expression \( 2^3 \), 2 is the base, and 3 is the exponent, meaning 2 is multiplied by itself three times: \( 2 \times 2 \times 2 = 8 \).
Exponents can also be fractional, allowing us to represent roots as powers. A common example is \( x^{1/2} \), which is the square root of \( x \). Understanding exponents is crucial when dealing with radicals because they provide an alternative way to express them and make calculations more manageable.

Radical Form

Radicals are a way to express roots of numbers. A radical expression includes the radical symbol (\( \sqrt{} \)) and the radicand, which is the number inside the symbol. For instance, \( \sqrt{a} \) represents the square root of \( a \).
When working with higher roots, such as cube roots or fourth roots, the root is indicated as an index outside the radical: \( \sqrt[3]{b} \) or \( \sqrt[4]{c} \).
Every radical can also be expressed using exponents. For example, \( \sqrt[4]{2} \) is another way to write \( 2^{1/4} \). This conversion helps simplify radical expressions and is particularly useful in multiplication and division operations.

Fractions

Fractions represent parts of a whole and are written as \( \frac{a}{b} \), where \( a \) is the numerator, and \( b \) is the denominator. In working with exponents, fractions often appear when dealing with roots.
To add or subtract fractions, it's necessary to have a common denominator. For instance, \( \frac{1}{2} \) and \( \frac{1}{4} \) can be added by expressing \( \frac{1}{2} \) as \( \frac{2}{4} \), making it easy to compute:

• Convert \( \frac{1}{2} \) to \( \frac{2}{4} \).
• Then, \( \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \).

This common denominator process is vital for simplifying operations in both fractional exponents and standard fraction arithmetic.

Multiplying Exponents

When multiplying numbers with the same base but different exponents, we add the exponents together. This property simplifies calculations significantly. For example, multiplying \( 2^{1/2} \) and \( 2^{1/4} \) results in:

• The base remains the same (i.e., 2).
• The exponents are added together: \( 1/2 + 1/4 \).
• To add \( 1/2 \) and \( 1/4 \), convert them to have a common denominator: \( 1/2 = 2/4 \), so \( 2/4 + 1/4 = 3/4 \).

Thus, the result of multiplying \( 2^{1/2} \times 2^{1/4} \) is \( 2^{3/4} \). Using this property helps to reach the simplest radical form efficiently.