Problem 31 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[4]{\frac{3}{4}} \cdot \sqrt[4]{\frac{27}{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{2} \).

Step 1: Break Down the Expression

The expression involves two fourth roots being multiplied together: \( \sqrt[4]{\frac{3}{4}} \cdot \sqrt[4]{\frac{27}{4}} \). To simplify, we will first use the property of roots, which allows us to combine these under the same root.

Step 2: Combine Under a Single Root

Using the property \( \sqrt[4]{a} \cdot \sqrt[4]{b} = \sqrt[4]{a \times b} \), combine the expressions: \( \sqrt[4]{\frac{3}{4} \times \frac{27}{4}} \).

Step 3: Multiply the Fractions

Multiply the fractions inside the root: \( \frac{3}{4} \times \frac{27}{4} = \frac{3 \times 27}{4 \times 4} = \frac{81}{16} \). The expression becomes \( \sqrt[4]{\frac{81}{16}} \).

Step 4: Simplify the Fourth Root

The fourth root of a fraction can be simplified by taking the fourth root of the numerator and the fourth root of the denominator separately. Calculate \( \sqrt[4]{81} \) and \( \sqrt[4]{16} \).

Step 5: Calculate Individual Fourth Roots

Since \( 81 = 3^4 \) and \( 16 = 2^4 \), it follows that \( \sqrt[4]{81} = 3 \) and \( \sqrt[4]{16} = 2 \).

Step 6: Find the Simplified Expression

Combine the results to find the simplified expression: \( \frac{3}{2} \).

Key Concepts

Roots and Radicals

Roots and radicals allow us to find the original value when an expression has been raised to a power. In particular, the task in our exercise is to deal with fourth roots. The fourth root, denoted as \( \sqrt[4]{x} \), indicates a number that, when multiplied by itself four times, equals \( x \). This principle helps us simplify expressions involving roots by breaking them down into more manageable numbers.

• Fourth roots are written as \(\sqrt[4]{x}\), which means the expression is seeking a base number that returns \( x \) when raised to a power of four.
• Radicals can be combined under a single root if they share the same index. This property was used to join \( \sqrt[4]{\frac{3}{4}} \) and \( \sqrt[4]{\frac{27}{4}} \).

Understanding roots is essential for simplifying complex expressions, and this concept is a building block for further mathematical problem-solving.

Properties of Exponents

Exponents tell us how many times a number is multiplied by itself. Recognizing the relationship between roots and exponents is crucial. A fourth root is the inverse operation of raising a number to the fourth power. For this exercise, knowing that \( 81 = 3^4 \) and \( 16 = 2^4 \) helps us understand how to simplify the radicals.

• Exponents are essential in determining the fourth powers of numbers, which helps to simplify expressions like \( \sqrt[4]{81} \) to 3 easily.
• This requires breaking numbers into their prime factors and seeing links with powers of small integers. For instance, 81 as \( 3^4 \) and 16 as \( 2^4 \).

Recognizing these exponent properties gives us the tools to transform complicated expressions into simpler forms by identifying such relationships.

Fraction Operations

Fractions represent parts of a whole and involve operations such as multiplication and simplification. The manipulation of fractions under a root means handling both multiplication of fractions and the simplification of their roots individually. To multiply fractions, as done in our example, multiply the numerators and the denominators separately: \( \frac{3}{4} \times \frac{27}{4} = \frac{81}{16} \).

• The product of two fractions \( \frac{a}{b} \times \frac{c}{d} \) results in \( \frac{a \times c}{b \times d} \).
• Simplification involves dividing both the numerator and the denominator by their greatest common divisor, if applicable.

In this exercise, understanding fraction operations allows us to manage expressions within and outside of radicals effectively, a necessary skill for this type of problem-solving.