Simplify $ \frac{12 Y^7}{18 Y^{-3}} $ Assume $ Y \neq 0 $A. $ \frac{y^{10}}{6} $B. $ \frac{2}{3 Y^{10}} $C. $ \frac{2y^{10}}{3} $

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Simplify $ \frac{12 y^7}{18 y^{-3}} $ Assume $ y \neq 0 $

Understanding the Problem

When simplifying a fraction, we need to reduce it to its simplest form by canceling out any common factors in the numerator and denominator. In this case, we have a fraction with a variable yy in both the numerator and denominator. Our goal is to simplify this fraction and express it in a more manageable form.

Step 1: Simplify the Numerator and Denominator

To simplify the fraction, we need to simplify the numerator and denominator separately. The numerator is 12y712y^7, and the denominator is 18y−318y^{-3}. We can simplify the numerator by factoring out the greatest common factor (GCF) of the coefficients, which is 6. We can also simplify the denominator by factoring out the GCF of the coefficients, which is 6.

Numerator: 12y^7 = 6(2y^7)
Denominator: 18y^{-3} = 6(3y^{-3})

Step 2: Cancel Out Common Factors

Now that we have simplified the numerator and denominator, we can cancel out any common factors. In this case, we can cancel out the common factor of 6 in both the numerator and denominator.

Simplified Fraction: (6(2y^7)) / (6(3y^{-3})) = (2y^7) / (3y^{-3})

Step 3: Simplify the Exponents

Now that we have canceled out the common factor of 6, we can simplify the exponents in the numerator and denominator. In this case, we can simplify the exponent of yy in the numerator by adding 7 to the exponent of yy in the denominator.

Simplified Fraction: (2y^7) / (3y^{-3}) = 2y^(7-(-3)) = 2y^(7+3) = 2y^10

Step 4: Simplify the Fraction

Now that we have simplified the exponents, we can simplify the fraction by dividing the numerator by the denominator. In this case, we can simplify the fraction by dividing 2 by 3.

Simplified Fraction: 2y^10 / 3

Conclusion

In conclusion, we have simplified the fraction $ \frac{12 y^7}{18 y^{-3}} $ by canceling out common factors and simplifying the exponents. The simplified fraction is $ \frac{2y^{10}}{3} $.

Answer

The correct answer is C. $ \frac{2y^{10}}{3} $.

Discussion

This problem requires the application of algebraic rules and techniques to simplify a fraction with a variable in both the numerator and denominator. The key steps involved in simplifying this fraction are canceling out common factors, simplifying the exponents, and dividing the numerator by the denominator. This problem is relevant to the topic of algebra and requires the use of algebraic rules and techniques to solve.

Related Problems

  • Simplify the fraction $ \frac{15x4}{20x{-2}} $
  • Simplify the fraction $ \frac{24y3}{32y{-1}} $
  • Simplify the fraction $ \frac{36z2}{48z{-4}} $

Practice Problems

  • Simplify the fraction $ \frac{12a5}{18a{-2}} $
  • Simplify the fraction $ \frac{20b3}{25b{-1}} $
  • Simplify the fraction $ \frac{30c2}{35c{-4}} $

Solutions

  • Simplify the fraction $ \frac12a5}{18a{-2}} $ $ \frac{2a^7{3} $
  • Simplify the fraction $ \frac20b3}{25b{-1}} $ $ \frac{4b^4{5} $
  • Simplify the fraction $ \frac30c2}{35c{-4}} $ $ \frac{6c^6{7} $

Conclusion

In conclusion, we have simplified the fraction $ \frac{12 y^7}{18 y^{-3}} $ by canceling out common factors and simplifying the exponents. The simplified fraction is $ \frac{2y^{10}}{3} $. This problem requires the application of algebraic rules and techniques to simplify a fraction with a variable in both the numerator and denominator. The key steps involved in simplifying this fraction are canceling out common factors, simplifying the exponents, and dividing the numerator by the denominator. This problem is relevant to the topic of algebra and requires the use of algebraic rules and techniques to solve.
Simplify $ \frac{12 y^7}{18 y^{-3}} $ Assume $ y \neq 0 $: Q&A

Q: What is the simplified form of the fraction $ \frac{12 y^7}{18 y^{-3}} $?

A: The simplified form of the fraction $ \frac{12 y^7}{18 y^{-3}} $ is $ \frac{2y^{10}}{3} $.

Q: How do I simplify the fraction $ \frac{12 y^7}{18 y^{-3}} $?

A: To simplify the fraction $ \frac{12 y^7}{18 y^{-3}} $, you need to cancel out common factors and simplify the exponents. First, simplify the numerator and denominator separately by factoring out the greatest common factor (GCF) of the coefficients. Then, cancel out any common factors and simplify the exponents.

Q: What is the greatest common factor (GCF) of the coefficients in the numerator and denominator?

A: The greatest common factor (GCF) of the coefficients in the numerator and denominator is 6.

Q: How do I simplify the exponents in the numerator and denominator?

A: To simplify the exponents in the numerator and denominator, you need to add or subtract the exponents. In this case, you can simplify the exponent of yy in the numerator by adding 7 to the exponent of yy in the denominator.

Q: What is the simplified form of the fraction after simplifying the exponents?

A: The simplified form of the fraction after simplifying the exponents is 2y102y^{10}.

Q: How do I simplify the fraction $ \frac{2y^{10}}{3} $?

A: To simplify the fraction $ \frac{2y^{10}}{3} $, you can divide the numerator by the denominator.

Q: What is the final simplified form of the fraction?

A: The final simplified form of the fraction is $ \frac{2y^{10}}{3} $.

Q: What are some related problems that I can try to practice my skills?

A: Some related problems that you can try to practice your skills include:

  • Simplify the fraction $ \frac{15x4}{20x{-2}} $
  • Simplify the fraction $ \frac{24y3}{32y{-1}} $
  • Simplify the fraction $ \frac{36z2}{48z{-4}} $

Q: What are some practice problems that I can try to test my skills?

A: Some practice problems that you can try to test your skills include:

  • Simplify the fraction $ \frac{12a5}{18a{-2}} $
  • Simplify the fraction $ \frac{20b3}{25b{-1}} $
  • Simplify the fraction $ \frac{30c2}{35c{-4}} $

Q: What are the solutions to the practice problems?

A: The solutions to the practice problems are:

  • Simplify the fraction $ \frac12a5}{18a{-2}} $ $ \frac{2a^7{3} $
  • Simplify the fraction $ \frac20b3}{25b{-1}} $ $ \frac{4b^4{5} $
  • Simplify the fraction $ \frac30c2}{35c{-4}} $ $ \frac{6c^6{7} $

Conclusion

In conclusion, we have simplified the fraction $ \frac{12 y^7}{18 y^{-3}} $ by canceling out common factors and simplifying the exponents. The simplified fraction is $ \frac{2y^{10}}{3} $. This problem requires the application of algebraic rules and techniques to simplify a fraction with a variable in both the numerator and denominator. The key steps involved in simplifying this fraction are canceling out common factors, simplifying the exponents, and dividing the numerator by the denominator. This problem is relevant to the topic of algebra and requires the use of algebraic rules and techniques to solve.