Fully Simplify $\frac{15 C^3}{25 C^9}$.

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Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves reducing an expression to its simplest form, making it easier to work with and understand. In this article, we will focus on simplifying the expression $\frac{15 c^3}{25 c^9}$, which is a great example of how to simplify complex algebraic expressions.

Understanding the Expression

The given expression is a fraction, where the numerator is $15 c^3$ and the denominator is $25 c^9$. To simplify this expression, we need to understand the properties of exponents and fractions.

Properties of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $c^3$ means $c \times c \times c$. When we multiply two numbers with the same base, we add their exponents. For example, $c^3 \times c^4 = c^{3+4} = c^7$.

Properties of Fractions

Fractions are a way of representing a part of a whole. When we divide two numbers, we can represent the result as a fraction. For example, $\frac{15}{25}$ means $15$ divided by $25$.

Simplifying the Expression

Now that we understand the properties of exponents and fractions, we can simplify the expression $\frac{15 c^3}{25 c^9}$.

Step 1: Factor Out Common Terms

The first step in simplifying the expression is to factor out common terms. In this case, we can factor out $5$ from both the numerator and the denominator.

15c325c9=5×3c35×5c9\frac{15 c^3}{25 c^9} = \frac{5 \times 3 c^3}{5 \times 5 c^9}

Step 2: Cancel Out Common Factors

Now that we have factored out common terms, we can cancel out common factors. In this case, we can cancel out $5$ from both the numerator and the denominator.

5×3c35×5c9=3c35c9\frac{5 \times 3 c^3}{5 \times 5 c^9} = \frac{3 c^3}{5 c^9}

Step 3: Simplify the Exponents

Now that we have cancelled out common factors, we can simplify the exponents. In this case, we can subtract the exponent of the denominator from the exponent of the numerator.

3c35c9=3c3−95c9=3c−65c9\frac{3 c^3}{5 c^9} = \frac{3 c^{3-9}}{5 c^9} = \frac{3 c^{-6}}{5 c^9}

Step 4: Simplify the Fraction

Finally, we can simplify the fraction by dividing the numerator by the denominator.

3c−65c9=35c9+6=35c15\frac{3 c^{-6}}{5 c^9} = \frac{3}{5 c^{9+6}} = \frac{3}{5 c^{15}}

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics. By understanding the properties of exponents and fractions, we can simplify expressions like $\frac{15 c^3}{25 c^9}$. In this article, we have shown how to simplify this expression by factoring out common terms, cancelling out common factors, simplifying the exponents, and simplifying the fraction.

Tips and Tricks

  • When simplifying complex algebraic expressions, always start by factoring out common terms.
  • When cancelling out common factors, make sure to cancel out the same number of factors from both the numerator and the denominator.
  • When simplifying the exponents, always subtract the exponent of the denominator from the exponent of the numerator.
  • When simplifying the fraction, always divide the numerator by the denominator.

Examples

Here are some examples of simplifying complex algebraic expressions:

  • 12x418x8=2x43x8=23x4\frac{12 x^4}{18 x^8} = \frac{2 x^4}{3 x^8} = \frac{2}{3 x^4}

  • 20y330y9=2y33y9=23y6\frac{20 y^3}{30 y^9} = \frac{2 y^3}{3 y^9} = \frac{2}{3 y^6}

  • 15z225z6=3z25z6=35z4\frac{15 z^2}{25 z^6} = \frac{3 z^2}{5 z^6} = \frac{3}{5 z^4}

Practice Problems

Here are some practice problems to help you practice simplifying complex algebraic expressions:

  • 24a536a9\frac{24 a^5}{36 a^9}

  • 30b340b7\frac{30 b^3}{40 b^7}

  • 18c224c6\frac{18 c^2}{24 c^6}

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics. By understanding the properties of exponents and fractions, we can simplify expressions like $\frac{15 c^3}{25 c^9}$. In this article, we have shown how to simplify this expression by factoring out common terms, cancelling out common factors, simplifying the exponents, and simplifying the fraction. We have also provided some tips and tricks, examples, and practice problems to help you practice simplifying complex algebraic expressions.

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Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. In our previous article, we provided a step-by-step guide on how to simplify the expression $\frac{15 c^3}{25 c^9}$. In this article, we will provide a Q&A guide to help you understand and practice simplifying complex algebraic expressions.

Q&A

Q: What is the first step in simplifying a complex algebraic expression?

A: The first step in simplifying a complex algebraic expression is to factor out common terms.

Q: How do I factor out common terms?

A: To factor out common terms, look for the greatest common factor (GCF) of the numerator and the denominator. Then, divide both the numerator and the denominator by the GCF.

Q: What is the next step after factoring out common terms?

A: After factoring out common terms, cancel out common factors. This involves dividing both the numerator and the denominator by the common factor.

Q: How do I simplify the exponents?

A: To simplify the exponents, subtract the exponent of the denominator from the exponent of the numerator.

Q: What is the final step in simplifying a complex algebraic expression?

A: The final step in simplifying a complex algebraic expression is to simplify the fraction. This involves dividing the numerator by the denominator.

Q: What are some common mistakes to avoid when simplifying complex algebraic expressions?

A: Some common mistakes to avoid when simplifying complex algebraic expressions include:

  • Not factoring out common terms
  • Not cancelling out common factors
  • Not simplifying the exponents
  • Not simplifying the fraction

Q: How can I practice simplifying complex algebraic expressions?

A: You can practice simplifying complex algebraic expressions by working through examples and practice problems. You can also use online resources and math software to help you practice.

Examples

Here are some examples of simplifying complex algebraic expressions:

  • 12x418x8=2x43x8=23x4\frac{12 x^4}{18 x^8} = \frac{2 x^4}{3 x^8} = \frac{2}{3 x^4}

  • 20y330y9=2y33y9=23y6\frac{20 y^3}{30 y^9} = \frac{2 y^3}{3 y^9} = \frac{2}{3 y^6}

  • 15z225z6=3z25z6=35z4\frac{15 z^2}{25 z^6} = \frac{3 z^2}{5 z^6} = \frac{3}{5 z^4}

Practice Problems

Here are some practice problems to help you practice simplifying complex algebraic expressions:

  • 24a536a9\frac{24 a^5}{36 a^9}

  • 30b340b7\frac{30 b^3}{40 b^7}

  • 18c224c6\frac{18 c^2}{24 c^6}

Tips and Tricks

  • When simplifying complex algebraic expressions, always start by factoring out common terms.
  • When cancelling out common factors, make sure to cancel out the same number of factors from both the numerator and the denominator.
  • When simplifying the exponents, always subtract the exponent of the denominator from the exponent of the numerator.
  • When simplifying the fraction, always divide the numerator by the denominator.

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics. By understanding the properties of exponents and fractions, we can simplify expressions like $\frac{15 c^3}{25 c^9}$. In this article, we have provided a Q&A guide to help you understand and practice simplifying complex algebraic expressions. We have also provided some examples and practice problems to help you practice simplifying complex algebraic expressions.

Additional Resources

  • Khan Academy: Simplifying Algebraic Expressions
  • Mathway: Simplifying Algebraic Expressions
  • Wolfram Alpha: Simplifying Algebraic Expressions

Final Thoughts

Simplifying complex algebraic expressions is a crucial skill in mathematics. By practicing and mastering this skill, you can become a more confident and proficient math student. Remember to always start by factoring out common terms, cancelling out common factors, simplifying the exponents, and simplifying the fraction. With practice and patience, you can become a master of simplifying complex algebraic expressions.