Simplify The Expression:$\frac{12 R^3 S^5 - 18 R^2 S^2 + 3 R S}{3 R S}$

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Introduction

In this article, we will simplify the given expression: 12r3s5−18r2s2+3rs3rs\frac{12 r^3 s^5 - 18 r^2 s^2 + 3 r s}{3 r s}. This expression involves variables and constants, and we will use algebraic techniques to simplify it. We will break down the expression into smaller parts, factor out common terms, and cancel out any common factors in the numerator and denominator.

Step 1: Factor Out Common Terms

The first step in simplifying the expression is to factor out common terms from the numerator. We can start by factoring out the greatest common factor (GCF) of the three terms in the numerator.

12r3s5−18r2s2+3rs3rs\frac{12 r^3 s^5 - 18 r^2 s^2 + 3 r s}{3 r s}

We can see that the GCF of the three terms is 3rs3 r s. We can factor this out of each term in the numerator.

3rs(4r2s4−6rs+1)3rs\frac{3 r s (4 r^2 s^4 - 6 r s + 1)}{3 r s}

Step 2: Cancel Out Common Factors

Now that we have factored out the common term 3rs3 r s from the numerator, we can cancel it out with the same term in the denominator.

3rs(4r2s4−6rs+1)3rs\frac{3 r s (4 r^2 s^4 - 6 r s + 1)}{3 r s}

We can cancel out the 3rs3 r s in the numerator and denominator, leaving us with:

4r2s4−6rs+14 r^2 s^4 - 6 r s + 1

Step 3: Simplify the Expression

Now that we have simplified the expression, we can see that it is in the form of a quadratic expression. However, we can further simplify it by factoring out any common factors.

4r2s4−6rs+14 r^2 s^4 - 6 r s + 1

We can see that there are no common factors that can be factored out of this expression. Therefore, this is the simplified form of the given expression.

Conclusion

In this article, we simplified the given expression: 12r3s5−18r2s2+3rs3rs\frac{12 r^3 s^5 - 18 r^2 s^2 + 3 r s}{3 r s}. We broke down the expression into smaller parts, factored out common terms, and canceled out any common factors in the numerator and denominator. The simplified form of the expression is 4r2s4−6rs+14 r^2 s^4 - 6 r s + 1.

Final Answer

The final answer is: 4r2s4−6rs+1\boxed{4 r^2 s^4 - 6 r s + 1}

Additional Tips and Tricks

  • When simplifying expressions, it's essential to factor out common terms and cancel out any common factors in the numerator and denominator.
  • Use algebraic techniques such as factoring, canceling, and combining like terms to simplify expressions.
  • Pay attention to the order of operations and follow the correct order of operations when simplifying expressions.

Common Mistakes to Avoid

  • Failing to factor out common terms and cancel out common factors in the numerator and denominator.
  • Not following the correct order of operations when simplifying expressions.
  • Not combining like terms when simplifying expressions.

Real-World Applications

Simplifying expressions is a crucial skill in mathematics and has numerous real-world applications. Some examples include:

  • Simplifying algebraic expressions in physics and engineering to solve problems related to motion, energy, and forces.
  • Simplifying expressions in economics to analyze and model economic systems.
  • Simplifying expressions in computer science to write efficient algorithms and programs.

Conclusion

Introduction

In our previous article, we simplified the expression: 12r3s5−18r2s2+3rs3rs\frac{12 r^3 s^5 - 18 r^2 s^2 + 3 r s}{3 r s}. We broke down the expression into smaller parts, factored out common terms, and canceled out any common factors in the numerator and denominator. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to factor out common terms from the numerator. This involves identifying the greatest common factor (GCF) of the terms in the numerator and factoring it out.

Q: How do I identify the greatest common factor (GCF) of a set of terms?

A: To identify the GCF of a set of terms, look for the largest term that divides each of the terms in the set evenly. For example, if you have the terms 12, 18, and 24, the GCF is 6 because 6 divides each of these terms evenly.

Q: What is the difference between factoring and canceling?

A: Factoring involves breaking down an expression into smaller parts by identifying common factors. Canceling involves eliminating common factors between the numerator and denominator of an expression.

Q: Can I cancel out any common factors in the numerator and denominator?

A: Yes, you can cancel out any common factors in the numerator and denominator. However, make sure that the common factor is not a variable or a constant that is raised to a power.

Q: How do I know if an expression is simplified?

A: An expression is simplified when there are no common factors that can be canceled out between the numerator and denominator. Additionally, the expression should be in its simplest form, with no like terms that can be combined.

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

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

  • Failing to factor out common terms and cancel out common factors in the numerator and denominator.
  • Not following the correct order of operations when simplifying expressions.
  • Not combining like terms when simplifying expressions.

Q: How do I apply simplifying expressions in real-world scenarios?

A: Simplifying expressions is a crucial skill in mathematics that has numerous real-world applications. Some examples include:

  • Simplifying algebraic expressions in physics and engineering to solve problems related to motion, energy, and forces.
  • Simplifying expressions in economics to analyze and model economic systems.
  • Simplifying expressions in computer science to write efficient algorithms and programs.

Conclusion

In conclusion, simplifying expressions is a fundamental skill in mathematics that has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can simplify expressions and solve problems in various fields. Remember to factor out common terms, cancel out common factors, and combine like terms to simplify expressions.

Additional Tips and Tricks

  • Practice simplifying expressions regularly to develop your skills and build confidence.
  • Use algebraic techniques such as factoring, canceling, and combining like terms to simplify expressions.
  • Pay attention to the order of operations and follow the correct order of operations when simplifying expressions.

Real-World Applications

Simplifying expressions is a crucial skill in mathematics that has numerous real-world applications. Some examples include:

  • Simplifying algebraic expressions in physics and engineering to solve problems related to motion, energy, and forces.
  • Simplifying expressions in economics to analyze and model economic systems.
  • Simplifying expressions in computer science to write efficient algorithms and programs.

Conclusion

In conclusion, simplifying expressions is a fundamental skill in mathematics that has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can simplify expressions and solve problems in various fields. Remember to factor out common terms, cancel out common factors, and combine like terms to simplify expressions.