Simplify The Expression: 4 Y 2 ⋅ 3 X Y 3 ⋅ X 3 4y^2 \cdot 3xy^3 \cdot X^3 4 Y 2 ⋅ 3 X Y 3 ⋅ X 3

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Understanding the Problem

When simplifying an algebraic expression, we need to apply the rules of exponents and combine like terms. In this case, we are given the expression $4y^2 \cdot 3xy^3 \cdot x^3$ and we need to simplify it.

Applying the Rules of Exponents

To simplify the expression, we need to apply the rules of exponents. The rules of exponents state that when we multiply two or more variables with the same base, we add their exponents. In this case, we have three variables: $y$, $x$, and $x^3$. We can start by combining the variables with the same base.

Combining Like Terms

We can start by combining the variables with the same base. We have $y^2$ and $y^3$, which have the same base $y$. We can add their exponents to get $y^{2+3} = y^5$. Similarly, we have $x$ and $x^3$, which have the same base $x$. We can add their exponents to get $x^{1+3} = x^4$.

Simplifying the Expression

Now that we have combined the variables with the same base, we can simplify the expression. We have $4y^2 \cdot 3xy^3 \cdot x^3 = 4 \cdot 3 \cdot y^5 \cdot x^4$. We can simplify this expression by multiplying the coefficients and combining the variables.

Multiplying the Coefficients

We can multiply the coefficients $4$ and $3$ to get $12$. So, the expression becomes $12y^5 \cdot x^4$.

Combining the Variables

We can combine the variables $y^5$ and $x^4$ to get $12x^4y^5$.

Final Answer

The final answer is $12x^4y^5$.

Understanding the Concept of Exponents

Exponents are a fundamental concept in algebra, and they play a crucial role in simplifying expressions. When we have a variable raised to a power, we can multiply it by another variable with the same base to add their exponents. This is a key concept in simplifying expressions and solving equations.

Real-World Applications of Exponents

Exponents have many real-world applications, including finance, science, and engineering. In finance, exponents are used to calculate compound interest and investment returns. In science, exponents are used to describe the growth and decay of populations, chemical reactions, and physical systems. In engineering, exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

Tips for Simplifying Expressions

Simplifying expressions can be a challenging task, but there are some tips that can help. Here are some tips for simplifying expressions:

• Combine like terms: Combine variables with the same base to add their exponents.
• Apply the rules of exponents: Multiply variables with the same base to add their exponents.
• Simplify coefficients: Multiply coefficients to simplify the expression.
• Check for errors: Check the expression for errors and make sure that it is simplified correctly.

Conclusion

Simplifying expressions is an essential skill in algebra, and it requires a deep understanding of the rules of exponents and like terms. By combining like terms, applying the rules of exponents, and simplifying coefficients, we can simplify expressions and solve equations. Exponents have many real-world applications, and they play a crucial role in finance, science, and engineering. By following the tips for simplifying expressions, we can simplify expressions and solve equations with confidence.

Frequently Asked Questions

• What is the rule for combining like terms?: The rule for combining like terms is to add the exponents of variables with the same base.
• How do I apply the rules of exponents?: To apply the rules of exponents, multiply variables with the same base to add their exponents.
• What is the difference between a coefficient and a variable?: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value.
• How do I simplify an expression?: To simplify an expression, combine like terms, apply the rules of exponents, and simplify coefficients.

Additional Resources

• Algebra textbook: A comprehensive algebra textbook that covers the rules of exponents and like terms.
• Online resources: Online resources, such as Khan Academy and Mathway, that provide video lessons and interactive exercises on simplifying expressions.
• Practice problems: Practice problems that allow you to apply the rules of exponents and like terms to simplify expressions.
  

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Understanding the Problem

When simplifying an algebraic expression, we need to apply the rules of exponents and combine like terms. In this case, we are given the expression $4y^2 \cdot 3xy^3 \cdot x^3$ and we need to simplify it.

Q&A: Simplifying Expressions

Q: What is the rule for combining like terms?

A: The rule for combining like terms is to add the exponents of variables with the same base.

Q: How do I apply the rules of exponents?

A: To apply the rules of exponents, multiply variables with the same base to add their exponents.

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

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value.

Q: How do I simplify an expression?

A: To simplify an expression, combine like terms, apply the rules of exponents, and simplify coefficients.

Q: What is the final answer to the expression $4y^2 \cdot 3xy^3 \cdot x^3$?

A: The final answer is $12x^4y^5$.

Q: Can you explain the concept of exponents in more detail?

A: Exponents are a fundamental concept in algebra, and they play a crucial role in simplifying expressions. When we have a variable raised to a power, we can multiply it by another variable with the same base to add their exponents.

Q: How do I know when to use the rule of exponents?

A: You should use the rule of exponents when you have variables with the same base and you want to multiply them together.

Q: Can you provide some examples of real-world applications of exponents?

A: Exponents have many real-world applications, including finance, science, and engineering. In finance, exponents are used to calculate compound interest and investment returns. In science, exponents are used to describe the growth and decay of populations, chemical reactions, and physical systems. In engineering, exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: What are some tips for simplifying expressions?

A: Here are some tips for simplifying expressions:

• Combine like terms: Combine variables with the same base to add their exponents.
• Apply the rules of exponents: Multiply variables with the same base to add their exponents.
• Simplify coefficients: Multiply coefficients to simplify the expression.
• Check for errors: Check the expression for errors and make sure that it is simplified correctly.

Conclusion

Simplifying expressions is an essential skill in algebra, and it requires a deep understanding of the rules of exponents and like terms. By combining like terms, applying the rules of exponents, and simplifying coefficients, we can simplify expressions and solve equations. Exponents have many real-world applications, and they play a crucial role in finance, science, and engineering. By following the tips for simplifying expressions, we can simplify expressions and solve equations with confidence.

Frequently Asked Questions

• What is the rule for combining like terms?
• How do I apply the rules of exponents?
• What is the difference between a coefficient and a variable?
• How do I simplify an expression?
• What is the final answer to the expression $4y^2 \cdot 3xy^3 \cdot x^3$?
• Can you explain the concept of exponents in more detail?
• How do I know when to use the rule of exponents?
• Can you provide some examples of real-world applications of exponents?
• What are some tips for simplifying expressions?

Additional Resources

• Algebra textbook: A comprehensive algebra textbook that covers the rules of exponents and like terms.
• Online resources: Online resources, such as Khan Academy and Mathway, that provide video lessons and interactive exercises on simplifying expressions.
• Practice problems: Practice problems that allow you to apply the rules of exponents and like terms to simplify expressions.

Final Thoughts

Simplifying expressions is a crucial skill in algebra, and it requires a deep understanding of the rules of exponents and like terms. By combining like terms, applying the rules of exponents, and simplifying coefficients, we can simplify expressions and solve equations. Exponents have many real-world applications, and they play a crucial role in finance, science, and engineering. By following the tips for simplifying expressions, we can simplify expressions and solve equations with confidence.