Simplify The Expression:${ \frac{4y 2}{3y {-4}} }$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying the given expression $\frac{4y^2}{3y^{-4}}$. We will break down the steps involved in simplifying this expression and provide a clear explanation of the mathematical concepts used.

Understanding the Expression

The given expression is $\frac{4y^2}{3y^{-4}}$. To simplify this expression, we need to understand the properties of exponents and how to manipulate them. The expression consists of two terms: $4y^2$ and $3y^{-4}$. The first term is a product of a constant and a variable raised to a power, while the second term is a product of a constant and a variable raised to a negative power.

Simplifying Exponents

To simplify the expression, we need to apply the rules of exponents. The rule states that when we divide two terms with the same base, we subtract the exponents. In this case, we have $y^2$ and $y^{-4}$, which have the same base $y$. Therefore, we can subtract the exponents to get $y^{2-(-4)}$.

Applying the Rule of Exponents

Now, let's apply the rule of exponents to simplify the expression. We have $y^{2-(-4)}$, which simplifies to $y^{2+4}$. This is because subtracting a negative number is equivalent to adding a positive number.

Simplifying the Expression

Now that we have simplified the exponents, we can rewrite the expression as $\frac{4y^{2+4}}{3}$. This is because the constant $4$ is still multiplied by the variable $y$ raised to the power of $2+4$.

Applying the Rule of Quotient of Powers

The rule of quotient of powers states that when we divide two terms with the same base, we subtract the exponents. In this case, we have $y^{2+4}$, which is equivalent to $y^6$. Therefore, we can rewrite the expression as $\frac{4y^6}{3}$.

Final Simplification

Now that we have applied the rule of quotient of powers, we can simplify the expression further. We can rewrite the expression as $\frac{4}{3}y^6$. This is because the constant $4$ is still multiplied by the variable $y$ raised to the power of $6$.

Conclusion

In this article, we simplified the expression $\frac{4y^2}{3y^{-4}}$ using the rules of exponents and the rule of quotient of powers. We broke down the steps involved in simplifying the expression and provided a clear explanation of the mathematical concepts used. By following these steps, we were able to simplify the expression to $\frac{4}{3}y^6$.

Additional Tips and Tricks

• When simplifying expressions, it's essential to understand the properties of exponents and how to manipulate them.
• The rule of exponents states that when we divide two terms with the same base, we subtract the exponents.
• The rule of quotient of powers states that when we divide two terms with the same base, we subtract the exponents.
• When simplifying expressions, it's essential to apply the rules of exponents and the rule of quotient of powers in the correct order.

Frequently Asked Questions

• Q: What is the rule of exponents? A: The rule of exponents states that when we divide two terms with the same base, we subtract the exponents.
• Q: What is the rule of quotient of powers? A: The rule of quotient of powers states that when we divide two terms with the same base, we subtract the exponents.
• Q: How do I simplify expressions using the rules of exponents and the rule of quotient of powers? A: To simplify expressions using the rules of exponents and the rule of quotient of powers, you need to apply the rules in the correct order. First, apply the rule of exponents to simplify the exponents, and then apply the rule of quotient of powers to simplify the expression.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. By following the steps outlined in this article, you can simplify expressions using the rules of exponents and the rule of quotient of powers. Remember to apply the rules in the correct order and to understand the properties of exponents and how to manipulate them. With practice and patience, you can become proficient in simplifying algebraic expressions.

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Introduction

In our previous article, we simplified the expression $\frac{4y^2}{3y^{-4}}$ using the rules of exponents and the rule of quotient of powers. In this article, we will provide a Q&A section to help you better understand the concepts and techniques involved in simplifying algebraic expressions.

Q&A

Q: What is the rule of exponents?

A: The rule of exponents states that when we divide two terms with the same base, we subtract the exponents. For example, $\frac{y^m}{y^n} = y^{m-n}$.

Q: What is the rule of quotient of powers?

A: The rule of quotient of powers states that when we divide two terms with the same base, we subtract the exponents. For example, $\frac{y^m}{y^n} = y^{m-n}$.

Q: How do I simplify expressions using the rules of exponents and the rule of quotient of powers?

A: To simplify expressions using the rules of exponents and the rule of quotient of powers, you need to apply the rules in the correct order. First, apply the rule of exponents to simplify the exponents, and then apply the rule of quotient of powers to simplify the expression.

Q: What is the difference between $y^m$ and $y^{-m}$?

A: $y^m$ represents a positive exponent, while $y^{-m}$ represents a negative exponent. When we have a negative exponent, we can rewrite it as a positive exponent by flipping the fraction. For example, $y^{-m} = \frac{1}{y^m}$.

Q: How do I handle negative exponents?

A: When we have a negative exponent, we can rewrite it as a positive exponent by flipping the fraction. For example, $y^{-m} = \frac{1}{y^m}$.

Q: What is the order of operations for simplifying expressions?

A: The order of operations for simplifying expressions is:

1. Apply the rule of exponents to simplify the exponents.
2. Apply the rule of quotient of powers to simplify the expression.
3. Simplify any remaining fractions.

Q: How do I simplify expressions with multiple variables?

A: To simplify expressions with multiple variables, you need to apply the rules of exponents and the rule of quotient of powers separately for each variable. For example, $\frac{xy^m}{yz^n} = \frac{x}{y} \cdot y^{m-n}$.

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

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

• Not applying the rule of exponents correctly
• Not applying the rule of quotient of powers correctly
• Not simplifying fractions correctly
• Not handling negative exponents correctly

Conclusion

In this article, we provided a Q&A section to help you better understand the concepts and techniques involved in simplifying algebraic expressions. By following the rules of exponents and the rule of quotient of powers, you can simplify expressions and become proficient in algebra.

Additional Tips and Tricks

• Always apply the rule of exponents and the rule of quotient of powers in the correct order.
• Simplify fractions correctly by dividing the numerator and denominator by their greatest common divisor.
• Handle negative exponents correctly by rewriting them as positive exponents.
• Practice simplifying expressions with multiple variables to become proficient in algebra.

Frequently Asked Questions

• Q: What is the rule of exponents? A: The rule of exponents states that when we divide two terms with the same base, we subtract the exponents.
• Q: What is the rule of quotient of powers? A: The rule of quotient of powers states that when we divide two terms with the same base, we subtract the exponents.
• Q: How do I simplify expressions using the rules of exponents and the rule of quotient of powers? A: To simplify expressions using the rules of exponents and the rule of quotient of powers, you need to apply the rules in the correct order.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying expressions and excel in algebra.