Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 33
In Exercises \(5-48,\) simplify the given expressions. Express results with positive exponents only. $$\left(2 v^{2}\right)^{-6}$$
Short Answer
Expert verified
The simplified expression is \(\frac{1}{64v^{12}}\).
Step by step solution
01
Recognize the Negative Exponent
The negative exponent in the expression \((2v^{2})^{-6}\) implies taking the reciprocal of the base. This means we need to rewrite the expression as a fraction before simplifying.
02
Rewrite as the Reciprocal
Since the expression is \((2v^{2})^{-6}\), we rewrite it by taking the reciprocal: \(\frac{1}{(2v^{2})^{6}}\). This step changes the negative exponent to a positive exponent in the denominator.
03
Simplify the Exponent
Now, calculate \((2v^{2})^{6}\). This expression involves raising both 2 and \(v^{2}\) to the 6th power. According to the power of a product rule: \((ab)^{n} = a^{n}b^{n}\), apply it here for \((2v^{2})^{6}\).
04
Apply Power of a Product Rule
Raise each factor within the parenthesis to the 6th power: \((2)^{6}(v^{2})^{6}\). This becomes \(2^{6} \times (v^{2})^{6}\).
05
Calculate the Powers
Calculate \(2^{6}\) which is 64, and \((v^{2})^{6}\) which using the power of a power rule (\((x^{m})^{n} = x^{m\cdot n}\)) becomes \(v^{12}\). So, \((2v^{2})^{6} = 64v^{12}\).
06
Write the Final Simplified Expression
Incorporate the simplified denominator back into the fraction: \(\frac{1}{64v^{12}}\). This is the final result, written with positive exponents.
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.
Negative Exponent
Understanding negative exponents is essential in algebraic simplification. When you encounter a negative exponent, such as in the expression \((2v^2)^{-6}\), it signifies that you need to take the reciprocal of the base. Negative exponents transform numbers to their reciprocal which helps in dealing with divisions more conveniently.
For example:
For example:
- \(x^{-n} = \frac{1}{x^n}\)
Power of a Product Rule
The power of a product rule helps us simplify expressions where a product is raised to an exponent, like \((ab)^n = a^n b^n\). In our exercise, the expression \((2v^2)^6\) involves both a numerical factor (2) and a variable factor \(v^2\).
Applying the rule means:
Applying the rule means:
- Each part of the product inside the parenthesis is raised to the exponent separately.
- This turns \((2v^2)^6\) into \(2^6 \times (v^2)^6\).
Power of a Power Rule
The power of a power rule is another vital tool in algebraic simplification. It handles scenarios where an expression with an existing exponent is raised to another exponent. For instance, \((x^m)^n = x^{m \cdot n}\). This rule significantly reduces the complexity of expressions involving powers.
In our situation:
In our situation:
- Apply it to \((v^2)^6\), resulting in \(v^{2 \cdot 6} = v^{12}\).
Positive Exponents
Converting to positive exponents simplifies equations and is a common requirement for final presentation in algebra. Positive exponents indicate standard multiplication and are more intuitive to understand. They are considered 'cleaner' as they represent straightforward repetition.
In our exercise:
In our exercise:
- We began with the negative exponent \((2v^2)^{-6}\), recognized it as \(\frac{1}{(2v^2)^6}\), and simplified.
- The process involved converting negative exponents into positive by using reciprocals, as well as applying the power of product and power of power rules.
- The result, expressed entirely with positive exponents, is \(\frac{1}{64v^{12}}\).
