Simplify By Adding Like Terms. Write The Answer With All Variables In The Denominator.$\frac{3 X^{-2} X^3 Y}{y^{-4}} - 2xy^5$

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Understanding the Problem

In this problem, we are given an algebraic expression that involves variables and exponents. The expression is 3x−2x3yy−4−2xy5\frac{3 x^{-2} x^3 y}{y^{-4}} - 2xy^5. Our goal is to simplify this expression by adding like terms and writing the result with all variables in the denominator.

Step 1: Simplify the Expression

To simplify the expression, we need to start by simplifying the fraction 3x−2x3yy−4\frac{3 x^{-2} x^3 y}{y^{-4}}. We can do this by using the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. Applying this rule, we get:

3x−2x3yy−4=3x−2+3yy−4=3x1yy−4\frac{3 x^{-2} x^3 y}{y^{-4}} = \frac{3 x^{-2+3} y}{y^{-4}} = \frac{3 x^1 y}{y^{-4}}

Step 2: Simplify the Exponents

Next, we need to simplify the exponents in the expression. We can do this by using the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. Applying this rule, we get:

3x1yy−4=3x1y1y4=3x1y⋅y4=3x1y5\frac{3 x^1 y}{y^{-4}} = \frac{3 x^1 y}{\frac{1}{y^4}} = 3 x^1 y \cdot y^4 = 3 x^1 y^5

Step 3: Simplify the Expression Further

Now that we have simplified the fraction, we can simplify the expression further by combining like terms. We can do this by using the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. Applying this rule, we get:

3x1y5−2xy5=(3−2)x1y5=x1y53 x^1 y^5 - 2xy^5 = (3 - 2) x^1 y^5 = x^1 y^5

Step 4: Write the Result with All Variables in the Denominator

Finally, we need to write the result with all variables in the denominator. We can do this by using the rule of exponents that states 1am=a−m\frac{1}{a^m} = a^{-m}. Applying this rule, we get:

x1y5=y51x1=y5x−1x^1 y^5 = \frac{y^5}{\frac{1}{x^1}} = \frac{y^5}{x^{-1}}

Conclusion

In this problem, we simplified the expression 3x−2x3yy−4−2xy5\frac{3 x^{-2} x^3 y}{y^{-4}} - 2xy^5 by adding like terms and writing the result with all variables in the denominator. We started by simplifying the fraction, then simplified the exponents, combined like terms, and finally wrote the result with all variables in the denominator.

Key Takeaways

  • To simplify an expression, we need to start by simplifying the fraction.
  • We can simplify the exponents in an expression by using the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • We can combine like terms in an expression by using the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • We can write an expression with all variables in the denominator by using the rule of exponents that states 1am=a−m\frac{1}{a^m} = a^{-m}.

Practice Problems

  1. Simplify the expression 2x3y2x−2y−1−3xy3\frac{2 x^3 y^2}{x^{-2} y^{-1}} - 3xy^3.
  2. Simplify the expression 4x2y4x−1y−2−2xy2\frac{4 x^2 y^4}{x^{-1} y^{-2}} - 2xy^2.
  3. Simplify the expression 3x4y3x−3y−2−4xy4\frac{3 x^4 y^3}{x^{-3} y^{-2}} - 4xy^4.

Answer Key

  1. 2x5y3y−1−3xy3=2x5y41−3xy3=2x5y4−3xy3\frac{2 x^5 y^3}{y^{-1}} - 3xy^3 = \frac{2 x^5 y^4}{1} - 3xy^3 = 2 x^5 y^4 - 3xy^3
  2. 4x3y6y−2−2xy2=4x3y81−2xy2=4x3y8−2xy2\frac{4 x^3 y^6}{y^{-2}} - 2xy^2 = \frac{4 x^3 y^8}{1} - 2xy^2 = 4 x^3 y^8 - 2xy^2
  3. 3x7y5y−2−4xy4=3x7y71−4xy4=3x7y7−4xy4\frac{3 x^7 y^5}{y^{-2}} - 4xy^4 = \frac{3 x^7 y^7}{1} - 4xy^4 = 3 x^7 y^7 - 4xy^4
    Simplify by Adding Like Terms: Q&A =====================================

Q: What is the first step in simplifying an expression with variables and exponents?

A: The first step in simplifying an expression with variables and exponents is to simplify the fraction. This involves using the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n} to combine like terms.

Q: How do I simplify the exponents in an expression?

A: To simplify the exponents in an expression, you can use the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. This rule allows you to combine like terms and simplify the expression.

Q: What is the difference between a variable and an exponent?

A: A variable is a letter or symbol that represents a value, while an exponent is a power or a quantity that is raised to a power. For example, in the expression x2x^2, xx is the variable and 22 is the exponent.

Q: How do I combine like terms in an expression?

A: To combine like terms in an expression, you can use the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. This rule allows you to add or subtract like terms and simplify the expression.

Q: What is the final step in simplifying an expression with variables and exponents?

A: The final step in simplifying an expression with variables and exponents is to write the result with all variables in the denominator. This involves using the rule of exponents that states 1am=a−m\frac{1}{a^m} = a^{-m} to rewrite the expression.

Q: Can I simplify an expression with variables and exponents using a calculator?

A: Yes, you can simplify an expression with variables and exponents using a calculator. However, it's always a good idea to check your work by hand to make sure you understand the process.

Q: What are some common mistakes to avoid when simplifying expressions with variables and exponents?

A: Some common mistakes to avoid when simplifying expressions with variables and exponents include:

  • Forgetting to simplify the fraction
  • Not using the rule of exponents to combine like terms
  • Not writing the result with all variables in the denominator
  • Not checking your work by hand

Q: How can I practice simplifying expressions with variables and exponents?

A: You can practice simplifying expressions with variables and exponents by working through practice problems, such as those found in a math textbook or online resource. You can also try simplifying expressions on your own and checking your work with a calculator or a friend.

Q: What are some real-world applications of simplifying expressions with variables and exponents?

A: Simplifying expressions with variables and exponents has many real-world applications, including:

  • Calculating the area and perimeter of shapes
  • Determining the cost of goods and services
  • Understanding the behavior of physical systems
  • Making predictions and forecasts

Conclusion

Simplifying expressions with variables and exponents is an important skill that has many real-world applications. By following the steps outlined in this article, you can simplify expressions with variables and exponents and gain a deeper understanding of the underlying math. Remember to practice regularly and check your work to ensure accuracy.

Practice Problems

  1. Simplify the expression 2x3y2x−2y−1−3xy3\frac{2 x^3 y^2}{x^{-2} y^{-1}} - 3xy^3.
  2. Simplify the expression 4x2y4x−1y−2−2xy2\frac{4 x^2 y^4}{x^{-1} y^{-2}} - 2xy^2.
  3. Simplify the expression 3x4y3x−3y−2−4xy4\frac{3 x^4 y^3}{x^{-3} y^{-2}} - 4xy^4.

Answer Key

  1. 2x5y41−3xy3=2x5y4−3xy3\frac{2 x^5 y^4}{1} - 3xy^3 = 2 x^5 y^4 - 3xy^3
  2. 4x3y6y−2−2xy2=4x3y81−2xy2=4x3y8−2xy2\frac{4 x^3 y^6}{y^{-2}} - 2xy^2 = \frac{4 x^3 y^8}{1} - 2xy^2 = 4 x^3 y^8 - 2xy^2
  3. 3x7y5y−2−4xy4=3x7y71−4xy4=3x7y7−4xy4\frac{3 x^7 y^5}{y^{-2}} - 4xy^4 = \frac{3 x^7 y^7}{1} - 4xy^4 = 3 x^7 y^7 - 4xy^4