Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 42
Use a calculator to evaluate each expression. $$\left(\frac{4 a^{5 / 6} b^{-1 / 5}}{a^{2 / 3} b^{2}}\right)^{-1 / 2}$$
Short Answer
Expert verified
The simplified expression is \(\frac{b^{11/10}}{2a^{1/12}}\). With \(a=1\) and \(b=1\), it simplifies to \(\frac{1}{2}\).
Step by step solution
01
Simplify the Inner Expression - Simplifying the Bases
Rewrite the expression, separating the terms with the same bases. We have the expression \( \frac{4a^{5/6}b^{-1/5}}{a^{2/3}b^{2}} \). Simplify the expression by dealing with the terms with base \(a\) and \(b\). Divide the exponents for \(a\): \(a^{(5/6 - 2/3)} = a^{(5/6 - 4/6)} = a^{1/6}\). Divide the exponents for \(b\): \(b^{-1/5 - 2} = b^{-1/5 - 10/5} = b^{-11/5}\). Thus, the expression becomes \(4a^{1/6}b^{-11/5}\).
02
Apply the Negative Exponent
The entire expression \((4a^{1/6}b^{-11/5})\) is raised to the power of \(-1/2\). Apply the negative half exponent to each part: \((4)^{-1/2} \), \((a^{1/6})^{-1/2} = a^{-(1/6 \cdot 1/2)} = a^{-1/12}\), \((b^{-11/5})^{-1/2} = b^{11/10}\).
03
Calculate the Coefficient
Calculate \( (4)^{-1/2} \). This is equal to \( \frac{1}{4^{1/2}} \), which is \( \frac{1}{2} \), as \( 4^{1/2} \) is 2.
04
Combine the Simplified Expression
Combine the terms:The expression becomes \( \frac{1}{2} \cdot a^{-1/12} \cdot b^{11/10} \).Rewrite for clarity: This is equal to \( \frac{b^{11/10}}{2a^{1/12}} \).
05
Evaluate the Numerical Value Using a Calculator
Convert the expression into numerical form by assuming specific values for \(a\) and \(b\) to evaluate. For simplicity, assume \( a = 1 \) and \( b = 1 \) so \( \frac{1^{11/10}}{2 \cdot 1^{1/12}} = \frac{1}{2}\).
06
Summary: Result Interpretation
Substituting specific values, we found the expression yields \(\frac{1}{2}\). This represents a simplified numerical estimate for the expression under the given assumptions.
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 Exponents
When you see an exponent with a negative sign in expressions, it can initially seem confusing. However, understanding negative exponents is quite straightforward once you grasp the fundamental rule: a negative exponent signifies the reciprocal of the base raised to the corresponding positive exponent.
For example, consider the expression \( b^{-11/5} \). This can be rewritten as \( \frac{1}{b^{11/5}} \). Essentially, the negative sign "flips" the base to the denominator.
Here are some key things to remember about negative exponents:
For example, consider the expression \( b^{-11/5} \). This can be rewritten as \( \frac{1}{b^{11/5}} \). Essentially, the negative sign "flips" the base to the denominator.
Here are some key things to remember about negative exponents:
- The expression \( x^{-n} \) means \( \frac{1}{x^n} \).
- Negative exponents can be used to express division operations through multiplication by reciprocal.
- To simplify an expression with negative exponents, rewrite them as positive by adjusting their position in a fraction.
Exponent Rules
Exponent rules provide a foundation for simplifying expressions and are crucial when dealing with terms that have exponents. The main rules to remember include:
- Product of Powers Rule: When multiplying two expressions that have the same base, you add their exponents: \( x^a \cdot x^b = x^{a+b} \).
- Quotient of Powers Rule: When dividing two expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator: \( \frac{x^a}{x^b} = x^{a-b} \).
- Power of a Power Rule: When raising an expression with an exponent to another power, multiply the exponents: \( (x^a)^b = x^{a \cdot b} \).
- Power of a Product Rule: Apply the exponent to each part of a product: \( (xy)^a = x^a \cdot y^a \).
Calculator Usage
A calculator becomes an invaluable tool in simplifying expressions, especially when dealing with fractional or negative exponents. Here鈥檚 how you can make the most of it:
- Entering Exponents: Make sure you use the exponent feature of your calculator, often noted by a \(^\wedge\) or \(\text{EXP}\) button, to input exponents. Double-check your entries, especially with negative or fractional exponents.
- Dealing with Fractions: Convert fractions smoothly by using the division sign when entering a fractional exponent. Some calculators allow direct fractional input, enhancing accuracy.
- Switching Between Forms: For expressions involving negative exponents, you may need to convert them to positive by using the reciprocal function, if manually adjusting is cumbersome.
- Precision: Be aware of the calculator's precision settings. Ensure it displays results to a few significant figures to avoid rounding errors in complex calculations.
