Which Expression Is Equivalent To $\left(\frac{a^{-4} B}{a^{-5} B^3}\right$\]?Assume $a \neq 0$, $b \neq 0$.A. $a^9 B^6$B. $a^3 B^{12}$C. $\frac{1}{a^3 B^2}$D. $\frac{a^{33}}{b^6}$

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Introduction

When dealing with exponential expressions, it's essential to understand the rules of exponents to simplify complex expressions. In this article, we'll focus on simplifying the expression (aβˆ’4baβˆ’5b3)\left(\frac{a^{-4} b}{a^{-5} b^3}\right), assuming aβ‰ 0a \neq 0 and bβ‰ 0b \neq 0. We'll explore the properties of exponents and apply them to simplify the given expression.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, a3a^3 can be written as aβ‹…aβ‹…aa \cdot a \cdot a. When dealing with exponents, it's crucial to understand the rules of exponentiation, including the product rule, power rule, and quotient rule.

Product Rule

The product rule states that when multiplying two powers with the same base, we add the exponents. For example, a2β‹…a3=a2+3=a5a^2 \cdot a^3 = a^{2+3} = a^5.

Power Rule

The power rule states that when raising a power to another power, we multiply the exponents. For example, (a2)3=a2β‹…3=a6(a^2)^3 = a^{2 \cdot 3} = a^6.

Quotient Rule

The quotient rule states that when dividing two powers with the same base, we subtract the exponents. For example, a3a2=a3βˆ’2=a1=a\frac{a^3}{a^2} = a^{3-2} = a^1 = a.

Simplifying the Expression

Now that we've reviewed the rules of exponents, let's apply them to simplify the expression (aβˆ’4baβˆ’5b3)\left(\frac{a^{-4} b}{a^{-5} b^3}\right).

Step 1: Apply the Quotient Rule

To simplify the expression, we'll start by applying the quotient rule to the numerator and denominator. This will allow us to subtract the exponents.

(aβˆ’4baβˆ’5b3)=aβˆ’4baβˆ’5b3=aβˆ’4aβˆ’5β‹…bb3\left(\frac{a^{-4} b}{a^{-5} b^3}\right) = \frac{a^{-4} b}{a^{-5} b^3} = \frac{a^{-4}}{a^{-5}} \cdot \frac{b}{b^3}

Step 2: Simplify the Fraction

Now that we've applied the quotient rule, let's simplify the fraction.

aβˆ’4aβˆ’5=aβˆ’4βˆ’(βˆ’5)=a1=a\frac{a^{-4}}{a^{-5}} = a^{-4-(-5)} = a^{1} = a

bb3=b1βˆ’3=bβˆ’2=1b2\frac{b}{b^3} = b^{1-3} = b^{-2} = \frac{1}{b^2}

Step 3: Combine the Terms

Now that we've simplified the fraction, let's combine the terms.

(aβˆ’4baβˆ’5b3)=aβ‹…1b2=ab2\left(\frac{a^{-4} b}{a^{-5} b^3}\right) = a \cdot \frac{1}{b^2} = \frac{a}{b^2}

Conclusion

In conclusion, the expression (aβˆ’4baβˆ’5b3)\left(\frac{a^{-4} b}{a^{-5} b^3}\right) simplifies to ab2\frac{a}{b^2}. This is achieved by applying the quotient rule, simplifying the fraction, and combining the terms.

Answer

The correct answer is:

ab2\boxed{\frac{a}{b^2}}

This answer is equivalent to option C, 1a3b2\frac{1}{a^3 b^2}, but with the correct simplification.

Discussion

The expression (aβˆ’4baβˆ’5b3)\left(\frac{a^{-4} b}{a^{-5} b^3}\right) is a classic example of how to apply the quotient rule and simplify exponential expressions. By following the steps outlined in this article, you should be able to simplify complex expressions and arrive at the correct answer.

Final Thoughts

In conclusion, simplifying exponential expressions requires a deep understanding of the rules of exponents. By applying the product rule, power rule, and quotient rule, you can simplify complex expressions and arrive at the correct answer. Remember to always follow the order of operations and simplify the expression step-by-step.

Frequently Asked Questions

  • Q: What is the product rule? A: The product rule states that when multiplying two powers with the same base, we add the exponents.
  • Q: What is the power rule? A: The power rule states that when raising a power to another power, we multiply the exponents.
  • Q: What is the quotient rule? A: The quotient rule states that when dividing two powers with the same base, we subtract the exponents.

Additional Resources

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

Related Articles

  • Simplifying Exponential Expressions: A Step-by-Step Guide
  • Understanding Exponents: A Comprehensive Guide
  • Applying the Quotient Rule: A Step-by-Step Guide

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Introduction

Exponents and exponential functions are fundamental concepts in mathematics, and understanding them is crucial for success in various fields, including science, engineering, and finance. In this article, we'll address some of the most frequently asked questions about exponents and exponential functions.

Q&A

Q: What is an exponent?

A: An exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself.

Q: What is the difference between a power and an exponent?

A: A power is the result of raising a base number to a certain exponent. For example, a3a^3 is a power, and 3 is the exponent.

Q: What is the product rule for exponents?

A: The product rule states that when multiplying two powers with the same base, we add the exponents. For example, a2β‹…a3=a2+3=a5a^2 \cdot a^3 = a^{2+3} = a^5.

Q: What is the power rule for exponents?

A: The power rule states that when raising a power to another power, we multiply the exponents. For example, (a2)3=a2β‹…3=a6(a^2)^3 = a^{2 \cdot 3} = a^6.

Q: What is the quotient rule for exponents?

A: The quotient rule states that when dividing two powers with the same base, we subtract the exponents. For example, a3a2=a3βˆ’2=a1=a\frac{a^3}{a^2} = a^{3-2} = a^1 = a.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, follow these steps:

  1. Apply the quotient rule to simplify the fraction.
  2. Simplify the fraction by canceling out common factors.
  3. Combine the terms by adding or subtracting the exponents.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base number is being multiplied by itself a certain number of times. A negative exponent indicates that the base number is being divided by itself a certain number of times.

Q: How do I handle negative exponents?

A: To handle negative exponents, follow these steps:

  1. Rewrite the negative exponent as a positive exponent by flipping the fraction.
  2. Simplify the expression by applying the quotient rule.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.

Q: What is the one exponent rule?

A: The one exponent rule states that any number raised to the power of one is equal to itself. For example, a1=aa^1 = a.

Q: How do I evaluate an exponential expression with a variable base?

A: To evaluate an exponential expression with a variable base, follow these steps:

  1. Substitute the value of the variable into the expression.
  2. Simplify the expression by applying the rules of exponents.

Q: What is the difference between an exponential function and a polynomial function?

A: An exponential function is a function that involves an exponent, while a polynomial function is a function that involves only addition, subtraction, and multiplication of variables.

Conclusion

In conclusion, understanding exponents and exponential functions is crucial for success in various fields. By following the rules of exponents and simplifying exponential expressions, you can solve complex problems and arrive at the correct answer.

Final Thoughts

In conclusion, exponents and exponential functions are fundamental concepts in mathematics. By understanding the rules of exponents and simplifying exponential expressions, you can solve complex problems and arrive at the correct answer.

Frequently Asked Questions

  • Q: What is the product rule for exponents? A: The product rule states that when multiplying two powers with the same base, we add the exponents.
  • Q: What is the power rule for exponents? A: The power rule states that when raising a power to another power, we multiply the exponents.
  • Q: What is the quotient rule for exponents? A: The quotient rule states that when dividing two powers with the same base, we subtract the exponents.

Additional Resources

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

Related Articles

  • Simplifying Exponential Expressions: A Step-by-Step Guide
  • Understanding Exponents: A Comprehensive Guide
  • Applying the Quotient Rule: A Step-by-Step Guide