Simplify The Expression: $2xy^3 \cdot 2x^{-3}$

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Understanding the Problem

When simplifying an expression, we need to apply the rules of exponents and multiplication. In this case, we are given the expression $2xy^3 \cdot 2x^{-3}$, and we need to simplify it.

The Rules of Exponents

Before we start simplifying the expression, let's review the rules of exponents. When multiplying two numbers with the same base, we add their exponents. For example, $x^a \cdot x^b = x^{a+b}$. However, when multiplying two numbers with different bases, we multiply the numbers and keep the exponents as they are.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the expression $2xy^3 \cdot 2x^{-3}$. We can start by multiplying the coefficients, which are the numbers in front of the variables.

Multiplying Coefficients

The coefficients are 2 and 2, so we multiply them together:

$$2 \cdot 2 = 4$$

So, the expression becomes $4xy^3 \cdot x^{-3}$.

Applying the Rules of Exponents

Now that we have multiplied the coefficients, we can apply the rules of exponents to simplify the expression. We have $x$ raised to the power of 1 and $x$ raised to the power of -3, so we add their exponents:

$$x^1 \cdot x^{-3} = x^{1+(-3)} = x^{-2}$$

So, the expression becomes $4x^{-2}y^3$.

Simplifying Negative Exponents

We can simplify negative exponents by moving the base to the other side of the fraction bar. In this case, we have $x^{-2}$, so we can rewrite it as $\frac{1}{x^2}$.

Final Simplified Expression

So, the final simplified expression is $\frac{4y^3}{x^2}$.

Conclusion

In this article, we have simplified the expression $2xy^3 \cdot 2x^{-3}$ using the rules of exponents and multiplication. We have reviewed the rules of exponents, applied them to the expression, and simplified the negative exponent. The final simplified expression is $\frac{4y^3}{x^2}$.

Common Mistakes to Avoid

When simplifying expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not applying the rules of exponents correctly: Make sure to add or subtract exponents correctly when multiplying or dividing variables with the same base.
• Not simplifying negative exponents: Negative exponents can be simplified by moving the base to the other side of the fraction bar.
• Not checking the final expression: Make sure to check the final expression to ensure that it is simplified correctly.

Practice Problems

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

• Simplify the expression $3x^2y \cdot 2x^{-1}$
• Simplify the expression $4y^3 \cdot 2y^{-2}$
• Simplify the expression $x^2y \cdot x^{-3}y^2$

Answer Key

Here are the answers to the practice problems:

• $6x^{2-1}y = 6xy$
• $8y^{3-2} = 8y$
• $x^{2-3}y^2 = \frac{y^2}{x}$
  Simplify the Expression: $2xy^3 \cdot 2x^{-3}$ - Q&A =====================================================

Understanding the Problem

When simplifying an expression, we need to apply the rules of exponents and multiplication. In this case, we are given the expression $2xy^3 \cdot 2x^{-3}$, and we need to simplify it.

Q&A

Q: What are the rules of exponents?

A: The rules of exponents state that when multiplying two numbers with the same base, we add their exponents. For example, $x^a \cdot x^b = x^{a+b}$. However, when multiplying two numbers with different bases, we multiply the numbers and keep the exponents as they are.

Q: How do I simplify negative exponents?

A: Negative exponents can be simplified by moving the base to the other side of the fraction bar. For example, $x^{-a} = \frac{1}{x^a}$.

Q: What is the final simplified expression for $2xy^3 \cdot 2x^{-3}$?

A: The final simplified expression is $\frac{4y^3}{x^2}$.

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

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

• Not applying the rules of exponents correctly
• Not simplifying negative exponents
• Not checking the final expression

Q: How do I practice simplifying expressions?

A: You can practice simplifying expressions by working through practice problems, such as:

• Simplify the expression $3x^2y \cdot 2x^{-1}$
• Simplify the expression $4y^3 \cdot 2y^{-2}$
• Simplify the expression $x^2y \cdot x^{-3}y^2$

Q: What are the answers to the practice problems?

A: The answers to the practice problems are:

• $6x^{2-1}y = 6xy$
• $8y^{3-2} = 8y$
• $x^{2-3}y^2 = \frac{y^2}{x}$

Additional Tips and Resources

Tips for Simplifying Expressions

• Make sure to apply the rules of exponents correctly
• Simplify negative exponents by moving the base to the other side of the fraction bar
• Check the final expression to ensure that it is simplified correctly

Resources for Learning More

• Khan Academy: Exponents and Exponential Functions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

Conclusion

In this article, we have simplified the expression $2xy^3 \cdot 2x^{-3}$ using the rules of exponents and multiplication. We have reviewed the rules of exponents, applied them to the expression, and simplified the negative exponent. The final simplified expression is $\frac{4y^3}{x^2}$. We have also provided a Q&A section to help you practice simplifying expressions and avoid common mistakes.