Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 29
Simplify. $$ \sqrt[5]{\frac{1}{32} w^{6} z^{7}} $$
Short Answer
Expert verified
\( \frac{w^{\frac{6}{5}} z^{\frac{7}{5}}}{2} \)
Step by step solution
01
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:\[ \sqrt[5]{2^{-5} \cdot w^6 \cdot z^7} \]
02
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:\[ \sqrt[5]{2^{-5}} \cdot \sqrt[5]{w^6} \cdot \sqrt[5]{z^7} \]
03
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. \( \sqrt[5]{z^7} = z^{\frac{7}{5}} \)
04
Combine the Simplified Terms
Combine the results from Step 3 to form the simplified expression:\[ \frac{1}{2} \cdot w^{\frac{6}{5}} \cdot z^{\frac{7}{5}} = \frac{w^{\frac{6}{5}} z^{\frac{7}{5}}}{2} \]
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
- 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 \( z^{\frac{7}{5}} \).
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.
