Fully Simplify The Expression.$-9x^5y(x^3y^3$\]Answer Attempt 1 Out Of 2:$\square$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves rewriting an expression in a more compact and manageable form, often by combining like terms or applying mathematical properties. In this article, we will focus on simplifying the given expression: −9x5y(x3y3)-9x^5y(x^3y^3). We will break down the process into manageable steps, using mathematical properties and techniques to arrive at the simplified expression.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its components. The expression consists of two terms: −9x5y-9x^5y and x3y3x^3y^3. The first term is a product of three factors: a negative coefficient, a variable raised to the power of 5, and another variable. The second term is also a product of two factors: a variable raised to the power of 3 and another variable.

Step 1: Distributing the Negative Coefficient

To simplify the expression, we can start by distributing the negative coefficient to the second term. This involves multiplying the negative coefficient by each factor in the second term. Using the distributive property, we can write:

−9x5y(x3y3)=−9x5y⋅x3y3-9x^5y(x^3y^3) = -9x^5y \cdot x^3y^3

Step 2: Combining Like Terms

Now that we have distributed the negative coefficient, we can combine like terms. In this case, we have two terms with the same variable raised to the same power: x5x^5 and x3x^3. We can combine these terms by adding their coefficients:

−9x5y⋅x3y3=−9x5y⋅x3y3=−9x8y4-9x^5y \cdot x^3y^3 = -9x^5y \cdot x^3y^3 = -9x^8y^4

Step 3: Simplifying the Expression

We have now simplified the expression by combining like terms and applying mathematical properties. The final simplified expression is:

−9x8y4-9x^8y^4

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it requires a deep understanding of mathematical properties and techniques. By following the steps outlined in this article, we have successfully simplified the given expression: −9x5y(x3y3)-9x^5y(x^3y^3). The final simplified expression is −9x8y4-9x^8y^4. We hope that this article has provided a clear and concise guide to simplifying algebraic expressions.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid. These include:

  • Not distributing the negative coefficient: Failing to distribute the negative coefficient can lead to incorrect simplifications.
  • Not combining like terms: Failing to combine like terms can result in an expression that is more complex than necessary.
  • Not applying mathematical properties: Failing to apply mathematical properties, such as the distributive property, can lead to incorrect simplifications.

Tips and Tricks

When simplifying algebraic expressions, there are several tips and tricks to keep in mind. These include:

  • Use the distributive property: The distributive property is a powerful tool for simplifying algebraic expressions. It allows us to distribute a coefficient to each factor in a term.
  • Combine like terms: Combining like terms is a crucial step in simplifying algebraic expressions. It involves adding the coefficients of terms with the same variable raised to the same power.
  • Apply mathematical properties: Mathematical properties, such as the commutative and associative properties, can be used to simplify algebraic expressions.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. These include:

  • Science and engineering: Simplifying algebraic expressions is essential in science and engineering, where complex mathematical models are used to describe physical systems.
  • Economics: Simplifying algebraic expressions is also important in economics, where mathematical models are used to describe economic systems.
  • Computer science: Simplifying algebraic expressions is a crucial step in computer science, where complex mathematical algorithms are used to solve problems.

Conclusion

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. In our previous article, we provided a step-by-step guide to simplifying the expression: −9x5y(x3y3)-9x^5y(x^3y^3). In this article, we will answer some common questions related to simplifying algebraic expressions.

Q: What is the difference between simplifying and evaluating an algebraic expression?

A: Simplifying an algebraic expression involves rewriting it in a more compact and manageable form, often by combining like terms or applying mathematical properties. Evaluating an algebraic expression, on the other hand, involves substituting specific values for the variables and calculating the resulting value.

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression whenever it is necessary to make the expression more manageable or to reveal its underlying structure. This can be the case when working with complex mathematical models, solving equations, or performing calculations.

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

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

  • Not distributing the negative coefficient
  • Not combining like terms
  • Not applying mathematical properties
  • Not checking for errors in the simplification process

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, you can use the following steps:

  1. Identify the variables and their exponents
  2. Combine like terms by adding or subtracting the coefficients of the variables with the same exponent
  3. Apply mathematical properties, such as the distributive property, to simplify the expression
  4. Check for errors in the simplification process

Q: Can I simplify an algebraic expression with a negative exponent?

A: Yes, you can simplify an algebraic expression with a negative exponent. To do this, you can use the following steps:

  1. Rewrite the expression with a positive exponent by taking the reciprocal of the base
  2. Simplify the expression using the rules for exponents
  3. Check for errors in the simplification process

Q: How do I simplify an algebraic expression with a fraction?

A: To simplify an algebraic expression with a fraction, you can use the following steps:

  1. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor
  2. Rewrite the expression with the simplified fraction
  3. Simplify the expression using the rules for fractions and exponents
  4. Check for errors in the simplification process

Q: Can I simplify an algebraic expression with a radical?

A: Yes, you can simplify an algebraic expression with a radical. To do this, you can use the following steps:

  1. Simplify the radical by finding the square root of the radicand
  2. Rewrite the expression with the simplified radical
  3. Simplify the expression using the rules for radicals and exponents
  4. Check for errors in the simplification process

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of mathematical properties and techniques. By following the steps outlined in this article, you can simplify algebraic expressions with multiple variables, negative exponents, fractions, and radicals. We hope that this article has provided a clear and concise guide to simplifying algebraic expressions.