Simplify ( R − 2 S 3 ) 2 ( 3 S ) 2 \left(r^{-2} S^3\right)^2(3 S)^2 ( R − 2 S 3 ) 2 ( 3 S ) 2 .A. 9 S 8 R 4 \frac{9 S^8}{r^4} R 4 9 S 8 ​ B. − 9 R 4 S 8 -9 R^4 S^8 − 9 R 4 S 8 C. 9 S 7 R 4 9 S^7 R^4 9 S 7 R 4 D. 3 S 7 1 4 \frac{3 S^7}{1^4} 1 4 3 S 7 ​

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Understanding the Problem

The given expression is (r−2s3)2(3s)2\left(r^{-2} s^3\right)^2(3 s)^2. To simplify this expression, we need to apply the rules of exponents and perform the necessary calculations.

Applying the Rules of Exponents

The first step is to expand the squared terms using the power rule of exponents, which states that (am)n=amn(a^m)^n = a^{mn}. Applying this rule to the given expression, we get:

(r−2s3)2=r−4s6\left(r^{-2} s^3\right)^2 = r^{-4} s^6

Expanding the Second Term

Next, we expand the second term (3s)2(3 s)^2 using the power rule of exponents:

(3s)2=32s2=9s2(3 s)^2 = 3^2 s^2 = 9 s^2

Combining the Terms

Now, we can combine the two simplified terms:

r−4s6⋅9s2=r−4s8⋅9r^{-4} s^6 \cdot 9 s^2 = r^{-4} s^8 \cdot 9

Simplifying the Expression

To simplify the expression further, we can rewrite it as a fraction with a negative exponent:

r−4s8⋅9=9s8r4r^{-4} s^8 \cdot 9 = \frac{9 s^8}{r^4}

Conclusion

Therefore, the simplified expression is 9s8r4\frac{9 s^8}{r^4}.

Answer Key

The correct answer is A. 9s8r4\frac{9 s^8}{r^4}.

Discussion

This problem requires the application of the rules of exponents, specifically the power rule, to simplify the given expression. The key concept is to expand the squared terms and then combine the resulting terms to obtain the final simplified expression.

Tips and Tricks

  • When simplifying expressions with exponents, it's essential to apply the rules of exponents correctly to avoid errors.
  • The power rule of exponents states that (am)n=amn(a^m)^n = a^{mn}, which can be applied to both positive and negative exponents.
  • When combining terms with exponents, it's crucial to multiply the exponents and add the bases.

Practice Problems

  • Simplify the expression (x2y3)2(2y)2\left(x^2 y^3\right)^2(2 y)^2.
  • Simplify the expression (a−3b4)2(3b)2\left(a^{-3} b^4\right)^2(3 b)^2.

Solutions

  • (x2y3)2(2y)2=x4y6⋅4y2=x4y8⋅4=4x4y814\left(x^2 y^3\right)^2(2 y)^2 = x^4 y^6 \cdot 4 y^2 = x^4 y^8 \cdot 4 = \frac{4 x^4 y^8}{1^4}
  • (a−3b4)2(3b)2=a−6b8⋅9b2=a−6b10⋅9=9b10a6\left(a^{-3} b^4\right)^2(3 b)^2 = a^{-6} b^8 \cdot 9 b^2 = a^{-6} b^{10} \cdot 9 = \frac{9 b^{10}}{a^6}

Q: What is the power rule of exponents?

A: The power rule of exponents states that (am)n=amn(a^m)^n = a^{mn}. This rule can be applied to both positive and negative exponents.

Q: How do I apply the power rule of exponents to the given expression?

A: To apply the power rule of exponents, you need to expand the squared terms using the rule (am)n=amn(a^m)^n = a^{mn}. In the given expression, (r−2s3)2\left(r^{-2} s^3\right)^2, you need to apply the power rule to get r−4s6r^{-4} s^6.

Q: What is the next step after expanding the first term?

A: After expanding the first term, you need to expand the second term (3s)2(3 s)^2 using the power rule of exponents. This will give you 9s29 s^2.

Q: How do I combine the two simplified terms?

A: To combine the two simplified terms, you need to multiply them together. In this case, you need to multiply r−4s6r^{-4} s^6 by 9s29 s^2.

Q: What is the final simplified expression?

A: The final simplified expression is 9s8r4\frac{9 s^8}{r^4}.

Q: What is the correct answer?

A: The correct answer is A. 9s8r4\frac{9 s^8}{r^4}.

Q: What are some tips and tricks for simplifying expressions with exponents?

A: Some tips and tricks for simplifying expressions with exponents include:

  • Applying the power rule of exponents correctly
  • Multiplying the exponents and adding the bases when combining terms
  • Using the rule (am)n=amn(a^m)^n = a^{mn} to expand squared terms

Q: What are some practice problems that I can try to reinforce my understanding of simplifying expressions with exponents?

A: Some practice problems that you can try to reinforce your understanding of simplifying expressions with exponents include:

  • Simplifying the expression (x2y3)2(2y)2\left(x^2 y^3\right)^2(2 y)^2
  • Simplifying the expression (a−3b4)2(3b)2\left(a^{-3} b^4\right)^2(3 b)^2

Q: What are the solutions to the practice problems?

A: The solutions to the practice problems are:

  • (x2y3)2(2y)2=x4y8⋅4=4x4y814\left(x^2 y^3\right)^2(2 y)^2 = x^4 y^8 \cdot 4 = \frac{4 x^4 y^8}{1^4}
  • (a−3b4)2(3b)2=a−6b10⋅9=9b10a6\left(a^{-3} b^4\right)^2(3 b)^2 = a^{-6} b^{10} \cdot 9 = \frac{9 b^{10}}{a^6}

Q: What are some common mistakes that I should avoid when simplifying expressions with exponents?

A: Some common mistakes that you should avoid when simplifying expressions with exponents include:

  • Not applying the power rule of exponents correctly
  • Not multiplying the exponents and adding the bases when combining terms
  • Not using the rule (am)n=amn(a^m)^n = a^{mn} to expand squared terms

Q: How can I check my work when simplifying expressions with exponents?

A: To check your work when simplifying expressions with exponents, you can:

  • Use the power rule of exponents to expand squared terms
  • Multiply the exponents and add the bases when combining terms
  • Use the rule (am)n=amn(a^m)^n = a^{mn} to expand squared terms
  • Check your work by plugging in values for the variables and simplifying the expression.