Simplify The Exponential Expression:$\[ \frac{4rs^{-3}}{\left(r^{-1}s^2\right)^3} \\]A. \[$\frac{4r^3}{s^{18}}\$\] B. \[$\frac{4r^4}{s^9}\$\] C. \[$\frac{4}{rs^8}\$\] D. \[$\frac{4}{rs^{10}}\$\]

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Understanding Exponential Expressions

Exponential expressions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we will focus on simplifying a given exponential expression, which involves applying the rules of exponents to arrive at a simplified form.

The Given Expression

The given expression is:

4rsβˆ’3(rβˆ’1s2)3{ \frac{4rs^{-3}}{\left(r^{-1}s^2\right)^3} }

To simplify this expression, we need to apply the rules of exponents, which include the product rule, power rule, and quotient rule.

Applying the Rules of Exponents

Product Rule

The product rule states that when we multiply two numbers with the same base, we add their exponents. However, in this case, we are dealing with a fraction, so we will apply the quotient rule instead.

Quotient Rule

The quotient rule states that when we divide two numbers with the same base, we subtract their exponents. In this case, we have:

4rsβˆ’3(rβˆ’1s2)3{ \frac{4rs^{-3}}{\left(r^{-1}s^2\right)^3} }

We can rewrite the denominator as:

(rβˆ’1s2)3=rβˆ’3s6{ \left(r^{-1}s^2\right)^3 = r^{-3}s^6 }

Now, we can apply the quotient rule:

4rsβˆ’3rβˆ’3s6=4r1βˆ’(βˆ’3)sβˆ’3βˆ’6{ \frac{4rs^{-3}}{r^{-3}s^6} = 4r^{1-(-3)}s^{-3-6} }

Simplifying further, we get:

4r4sβˆ’9{ 4r^4s^{-9} }

Power Rule

The power rule states that when we raise a power to a power, we multiply the exponents. In this case, we have:

4r4sβˆ’9{ 4r^4s^{-9} }

We can rewrite this as:

4r4s9{ \frac{4r^4}{s^9} }

Conclusion

In conclusion, the simplified form of the given exponential expression is:

4r4s9{ \frac{4r^4}{s^9} }

This is the correct answer among the options provided.

Final Answer

The final answer is:

4r4s9{ \boxed{\frac{4r^4}{s^9}} }

Discussion

This problem requires a good understanding of the rules of exponents, including the product rule, power rule, and quotient rule. It also requires the ability to apply these rules to simplify complex expressions.

Related Topics

  • Exponents and Powers
  • Rules of Exponents
  • Simplifying Expressions

Practice Problems

  • Simplify the expression: 3x2yβˆ’3(xβˆ’1y2)2{ \frac{3x^2y^{-3}}{\left(x^{-1}y^2\right)^2} }
  • Simplify the expression: 2a3bβˆ’2(aβˆ’2b3)2{ \frac{2a^3b^{-2}}{\left(a^{-2}b^3\right)^2} }

Solutions

  • The simplified form of the expression is: 3x5yβˆ’6xβˆ’2y4=3x7yβˆ’101=3x7y10{ \frac{3x^5y^{-6}}{x^{-2}y^4} = \frac{3x^7y^{-10}}{1} = \frac{3x^7}{y^{10}} }
  • The simplified form of the expression is: 2a5bβˆ’4aβˆ’4b6=2a9bβˆ’101=2a9b10{ \frac{2a^5b^{-4}}{a^{-4}b^6} = \frac{2a^9b^{-10}}{1} = \frac{2a^9}{b^{10}} }

Conclusion

In conclusion, simplifying exponential expressions requires a good understanding of the rules of exponents and the ability to apply these rules to simplify complex expressions. With practice and patience, you can become proficient in simplifying exponential expressions and solving mathematical problems with ease.

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Frequently Asked Questions

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that involves a base raised to a power. For example, 2^3 is an exponential expression where 2 is the base and 3 is the exponent.

Q: What are the rules of exponents?

A: The rules of exponents are a set of mathematical rules that govern the behavior of exponents in algebraic expressions. The three main rules of exponents are:

  • Product rule: a^m * a^n = a^(m+n)
  • Power rule: (am)n = a^(m*n)
  • Quotient rule: a^m / a^n = a^(m-n)

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you need to apply the rules of exponents. Here are the steps:

  1. Identify the base and exponent in the expression.
  2. Apply the product rule, power rule, or quotient rule as necessary.
  3. Simplify the expression by combining like terms.

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as a reciprocal. For example, 2^3 is equal to 222, while 2^-3 is equal to 1/2^3.

Q: How do I handle exponents with fractions?

A: When working with exponents and fractions, you need to apply the rules of exponents and fractions separately. For example, (1/2)^3 is equal to 1/2^3, which is equal to 1/8.

Q: Can I simplify an expression with multiple exponents?

A: Yes, you can simplify an expression with multiple exponents by applying the rules of exponents. For example, 2^3 * 2^4 can be simplified to 2^(3+4), which is equal to 2^7.

Q: How do I handle exponents with variables?

A: When working with exponents and variables, you need to apply the rules of exponents and variables separately. For example, x^2 * x^3 can be simplified to x^(2+3), which is equal to x^5.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by applying the rules of exponents. For example, 2^-3 can be simplified to 1/2^3, which is equal to 1/8.

Q: How do I handle exponents with parentheses?

A: When working with exponents and parentheses, you need to apply the rules of exponents and parentheses separately. For example, (23)4 can be simplified to 2^(3*4), which is equal to 2^12.

Common Mistakes

  • Forgetting to apply the product rule when multiplying exponents.
  • Forgetting to apply the power rule when raising an exponent to a power.
  • Forgetting to apply the quotient rule when dividing exponents.
  • Not simplifying the expression by combining like terms.

Tips and Tricks

  • Always apply the rules of exponents in the correct order.
  • Use parentheses to clarify the order of operations.
  • Simplify the expression by combining like terms.
  • Check your work by plugging in values for the variables.

Conclusion

In conclusion, simplifying exponential expressions requires a good understanding of the rules of exponents and the ability to apply these rules to simplify complex expressions. With practice and patience, you can become proficient in simplifying exponential expressions and solving mathematical problems with ease.