Simplify The Expression:${ \frac{6x^7 - 4x 4}{-2x 7} }$

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Introduction

Algebraic manipulation is a crucial skill in mathematics, and simplifying expressions is an essential part of it. In this article, we will focus on simplifying the given expression: 6x7−4x4−2x7\frac{6x^7 - 4x^4}{-2x^7}. We will break down the process into manageable steps, making it easier to understand and follow along.

Understanding the Expression

Before we start simplifying the expression, let's take a closer look at it. The given expression is a fraction, where the numerator is 6x7−4x46x^7 - 4x^4 and the denominator is −2x7-2x^7. Our goal is to simplify this expression, making it easier to work with.

Step 1: Factor Out Common Terms

To simplify the expression, we can start by factoring out common terms from the numerator. In this case, we can factor out 2x42x^4 from both terms in the numerator.

6x7−4x4−2x7=2x4(3x3−2)−2x7\frac{6x^7 - 4x^4}{-2x^7} = \frac{2x^4(3x^3 - 2)}{-2x^7}

Step 2: Simplify the Fraction

Now that we have factored out common terms, we can simplify the fraction. We can cancel out the common factor of 2x42x^4 from the numerator and denominator.

2x4(3x3−2)−2x7=3x3−2−x3\frac{2x^4(3x^3 - 2)}{-2x^7} = \frac{3x^3 - 2}{-x^3}

Step 3: Simplify the Expression Further

We can simplify the expression further by factoring out a negative sign from the denominator.

3x3−2−x3=−3x3−2x3\frac{3x^3 - 2}{-x^3} = -\frac{3x^3 - 2}{x^3}

Step 4: Final Simplification

Now that we have simplified the expression as much as possible, we can write the final answer.

−3x3−2x3=−3x3x3+2x3-\frac{3x^3 - 2}{x^3} = -\frac{3x^3}{x^3} + \frac{2}{x^3}

Step 5: Final Answer

Using the properties of exponents, we can simplify the expression further.

−3x3x3+2x3=−3+2x3-\frac{3x^3}{x^3} + \frac{2}{x^3} = -3 + \frac{2}{x^3}

Conclusion

In this article, we have simplified the given expression: 6x7−4x4−2x7\frac{6x^7 - 4x^4}{-2x^7}. We have broken down the process into manageable steps, making it easier to understand and follow along. By factoring out common terms, simplifying the fraction, and simplifying the expression further, we have arrived at the final answer: −3+2x3-3 + \frac{2}{x^3}.

Frequently Asked Questions

  • What is algebraic manipulation? Algebraic manipulation is the process of simplifying and rearranging algebraic expressions to make them easier to work with.
  • Why is simplifying expressions important? Simplifying expressions is important because it makes it easier to solve equations and inequalities, and it can also help to identify patterns and relationships between variables.
  • What are some common techniques used in algebraic manipulation? Some common techniques used in algebraic manipulation include factoring, simplifying fractions, and using the properties of exponents.

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires practice and patience to master. By following the steps outlined in this article, you can simplify even the most complex expressions and become more confident in your ability to work with algebraic expressions.

Additional Resources

  • Khan Academy: Algebraic Manipulation
  • Mathway: Algebraic Manipulation
  • Wolfram Alpha: Algebraic Manipulation

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "College Algebra" by James Stewart
  • "Algebra: A Comprehensive Introduction" by Christopher J. Rasmussen

Introduction

Algebraic manipulation is a crucial skill in mathematics, and it can be a challenging topic to master. In this article, we will answer some of the most frequently asked questions about algebraic manipulation, providing you with a better understanding of this important concept.

Q&A: Algebraic Manipulation

Q: What is algebraic manipulation?

A: Algebraic manipulation is the process of simplifying and rearranging algebraic expressions to make them easier to work with. This can involve factoring, simplifying fractions, and using the properties of exponents.

Q: Why is simplifying expressions important?

A: Simplifying expressions is important because it makes it easier to solve equations and inequalities, and it can also help to identify patterns and relationships between variables.

Q: What are some common techniques used in algebraic manipulation?

A: Some common techniques used in algebraic manipulation include factoring, simplifying fractions, and using the properties of exponents.

Q: How do I simplify a fraction?

A: To simplify a fraction, you can cancel out common factors between the numerator and denominator. For example, if you have the fraction 6x7−4x4−2x7\frac{6x^7 - 4x^4}{-2x^7}, you can factor out 2x42x^4 from the numerator and cancel it out with the denominator.

Q: What is factoring?

A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions. For example, the expression 6x7−4x46x^7 - 4x^4 can be factored as 2x4(3x3−2)2x^4(3x^3 - 2).

Q: How do I use the properties of exponents?

A: The properties of exponents state that when you multiply two powers with the same base, you can add their exponents. For example, x3⋅x2=x3+2=x5x^3 \cdot x^2 = x^{3+2} = x^5.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that remains the same.

Q: How do I solve an equation with variables on both sides?

A: To solve an equation with variables on both sides, you can use inverse operations to isolate the variable. For example, if you have the equation 2x+3=52x + 3 = 5, you can subtract 3 from both sides to get 2x=22x = 2, and then divide both sides by 2 to get x=1x = 1.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when working with expressions that involve multiple operations. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Conclusion

Algebraic manipulation is a crucial skill in mathematics, and it requires practice and patience to master. By understanding the techniques and concepts outlined in this article, you can become more confident in your ability to work with algebraic expressions and solve equations and inequalities.

Frequently Asked Questions

  • What is algebraic manipulation?
  • Why is simplifying expressions important?
  • What are some common techniques used in algebraic manipulation?
  • How do I simplify a fraction?
  • What is factoring?
  • How do I use the properties of exponents?
  • What is the difference between a variable and a constant?
  • How do I solve an equation with variables on both sides?
  • What is the order of operations?

Additional Resources

  • Khan Academy: Algebraic Manipulation
  • Mathway: Algebraic Manipulation
  • Wolfram Alpha: Algebraic Manipulation

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "College Algebra" by James Stewart
  • "Algebra: A Comprehensive Introduction" by Christopher J. Rasmussen