Problem 29 - Algebra 2

Question

Simplify. $$ \sqrt[5]{\frac{1}{32} w^{6} z^{7}} $$

Step-by-Step Solution

Answer

\( \frac{w^{\frac{6}{5}} z^{\frac{7}{5}}}{2} \)

Step 1: Rewrite Inside the Radical

First, recognize that the expression inside the 5th root can be written using a combination of integer powers. Notice that \( \frac{1}{32} \) is the same as \( 2^{-5} \) because 32 is \( 2^5 \). So, rewrite the expression as follows:\[ \sqrt[5]{2^{-5} \cdot w^6 \cdot z^7} \]

Step 2: Apply the 5th Root to Each Component

Since the 5th root of a product is equal to the product of the 5th roots, distribute the radical over each factor within the expression:\[ \sqrt[5]{2^{-5}} \cdot \sqrt[5]{w^6} \cdot \sqrt[5]{z^7} \]

Step 3: Simplify Each Component Separately

Simplify each fifth root individually:1. \( \sqrt[5]{2^{-5}} = 2^{-1} = \frac{1}{2} \)2. \( \sqrt[5]{w^6} = w^{\frac{6}{5}} \)3. \( \sqrt[5]{z^7} = z^{\frac{7}{5}} \)

Step 4: Combine the Simplified Terms

Combine the results from Step 3 to form the simplified expression:\[ \frac{1}{2} \cdot w^{\frac{6}{5}} \cdot z^{\frac{7}{5}} = \frac{w^{\frac{6}{5}} z^{\frac{7}{5}}}{2} \]

Key Concepts

Rational Exponents

Rational exponents are a way to express roots and powers using fractions. They are particularly useful when simplifying expressions with radicals.

• The base of the expression remains the same, and the exponent is written as a fraction.
• For example, the rational exponent notation for the nth root of a number is \( b^{\frac{1}{n}} \), where \( b \) is the base.

Imagine having a fifth root, like in the expression we are simplifying. Using rational exponents, \( \sqrt[5]{a} \) can be rewritten as \( a^{\frac{1}{5}} \). This transformation simplifies the manipulation of expressions, as combining and dividing exponents is easier than working directly with roots.Understanding rational exponents allows us to express roots of products and powers consistently, aiding in the simplification process.

nth Root

The nth root is a concept where you determine what number, when raised to an nth power, equals the original number. This concept is critical in understanding how to work with radical expressions.

• The nth root of a number \( a \) is represented by \( \sqrt[n]{a} \) and is equal to the number \( b \), such that \( b^n = a \).
• If the number has an nth root that is an integer, simplifying becomes straightforward.

In our example of simplifying \( \sqrt[5]{\frac{1}{32} w^{6} z^{7}} \), the 5th root transforms the expression by applying this root to each part separately.This step-by-step transformation allows the expression to be easier to simplify through the conversion to fractional exponents. Understanding this root operation is essential for managing complex expressions efficiently.

Fractional Exponents

Fractional exponents are a powerful tool in simplifying expressions that involve roots and powers. They provide an alternative format to radicals that is easier to manipulate mathematically.

• The numerator of a fractional exponent indicates the power to which the base is raised.
• The denominator represents the root that is being taken.

In our example, once the 5th root is accounted for, we convert to fractional exponents:

• The term \( \sqrt[5]{w^6} \) becomes \( w^{\frac{6}{5}} \), where 6 is the power and 5 is the root.
• Similarly, \( \sqrt[5]{z^7} \) turns into \( z^{\frac{7}{5}} \).

The use of fractional exponents simplifies expression manipulation and helps to reach the final, streamlined form more effectively.

Simplification Process

The simplification process involves shrinking complex mathematical expressions into simpler forms while retaining their value. It relies heavily on the rules of exponents and roots.

• Start by converting parts of the expression into more manageable terms, such as using rational exponents.
• Then, determine how to distribute operations, like roots, to each part of the expression.
• Simplify each component individually and then combine them back together.

In this exercise, the simplification was done by:1. Recognizing common factors, such as rewriting \( \frac{1}{32} \) as \( 2^{-5} \).2. Applying the 5th root separately to each element.3. Rewriting these roots with fractional exponents for ease of manipulation.4. Finally, combining the elements to reach the final form: \( \frac{w^{\frac{6}{5}} z^{\frac{7}{5}}}{2} \).By understanding and following the simplification process carefully, one can turn even intricate expressions into their most concise form.