Simplify The Expression: ${ \frac{\left(2 D^2 E\right)^2}{\left(a D^{-3} E {-2}\right) {-1}} }$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying a complex expression involving exponents and variables. The given expression is $\frac{\left(2 d^2 e\right)^2}{\left(a d^{-3} e^{-2}\right)^{-1}}$. Our goal is to simplify this expression by applying the rules of exponents and algebraic manipulation.

Understanding Exponents

Before we dive into simplifying the expression, let's review the rules of exponents. When we have a variable raised to a power, we can apply the following rules:

• Product of Powers Rule: When we multiply two variables with the same base, we add their exponents. For example, $x^a \cdot x^b = x^{a+b}$.
• Power of a Power Rule: When we raise a variable to a power and then raise it to another power, we multiply the exponents. For example, $(x^a)^b = x^{ab}$.
• Negative Exponent Rule: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $x^{-a} = \frac{1}{x^a}$.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's apply them to simplify the given expression. We can start by expanding the numerator and denominator using the product of powers rule.

$\frac{\left(2 d^2 e\right)^2}{\left(a d^{-3} e^{-2}\right)^{-1}} = \frac{2^2 \cdot d^4 \cdot e^2}{a \cdot d^{-3} \cdot e^{-2}}$

Next, we can apply the power of a power rule to simplify the numerator and denominator.

$\frac{2^2 \cdot d^4 \cdot e^2}{a \cdot d^{-3} \cdot e^{-2}} = \frac{4 \cdot d^4 \cdot e^2}{a \cdot d^{-3} \cdot e^{-2}}$

Now, we can apply the negative exponent rule to rewrite the denominator.

$\frac{4 \cdot d^4 \cdot e^2}{a \cdot d^{-3} \cdot e^{-2}} = \frac{4 \cdot d^4 \cdot e^2}{a} \cdot \frac{d^3}{e^2}$

Combining Like Terms

We can now combine like terms in the numerator and denominator.

$\frac{4 \cdot d^4 \cdot e^2}{a} \cdot \frac{d^3}{e^2} = \frac{4 \cdot d^7}{a}$

Final Simplification

We have now simplified the expression to its final form. The simplified expression is $\frac{4 \cdot d^7}{a}$.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we have focused on simplifying a complex expression involving exponents and variables. We have applied the rules of exponents and algebraic manipulation to simplify the expression to its final form. The simplified expression is $\frac{4 \cdot d^7}{a}$.

Frequently Asked Questions

• What is the product of powers rule? The product of powers rule states that when we multiply two variables with the same base, we add their exponents. For example, $x^a \cdot x^b = x^{a+b}$.
• What is the power of a power rule? The power of a power rule states that when we raise a variable to a power and then raise it to another power, we multiply the exponents. For example, $(x^a)^b = x^{ab}$.
• What is the negative exponent rule? The negative exponent rule states that when we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $x^{-a} = \frac{1}{x^a}$.

Further Reading

• Algebraic Manipulation: Algebraic manipulation is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we have focused on simplifying a complex expression involving exponents and variables.
• Exponents and Variables: Exponents and variables are fundamental concepts in mathematics, and they require a deep understanding of the underlying concepts. In this article, we have focused on simplifying a complex expression involving exponents and variables.
• Mathematics: Mathematics is a vast and fascinating field that encompasses a wide range of topics, including algebra, geometry, trigonometry, and calculus. In this article, we have focused on simplifying a complex expression involving exponents and variables.
  

Introduction

In our previous article, we simplified a complex expression involving exponents and variables. We applied the rules of exponents and algebraic manipulation to simplify the expression to its final form. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two variables with the same base, we add their exponents. For example, $x^a \cdot x^b = x^{a+b}$.

Q: What is the power of a power rule?

A: The power of a power rule states that when we raise a variable to a power and then raise it to another power, we multiply the exponents. For example, $(x^a)^b = x^{ab}$.

Q: What is the negative exponent rule?

A: The negative exponent rule states that when we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $x^{-a} = \frac{1}{x^a}$.

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

A: To simplify an expression with multiple variables and exponents, you can start by applying the product of powers rule, the power of a power rule, and the negative exponent rule. You can then combine like terms and simplify the expression further.

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

A: A variable is a symbol that represents a value that can change, while a constant is a value that remains the same. For example, $x$ is a variable, while $2$ is a constant.

Q: How do I apply the rules of exponents to simplify an expression?

A: To apply the rules of exponents to simplify an expression, you can start by identifying the base and the exponent. You can then apply the product of powers rule, the power of a power rule, and the negative exponent rule to simplify the expression.

Q: What is the final form of the expression $\frac{\left(2 d^2 e\right)^2}{\left(a d^{-3} e^{-2}\right)^{-1}}$?

A: The final form of the expression is $\frac{4 \cdot d^7}{a}$.

Conclusion

In this article, we have answered some frequently asked questions related to the simplification of the expression $\frac{\left(2 d^2 e\right)^2}{\left(a d^{-3} e^{-2}\right)^{-1}}$. We have applied the rules of exponents and algebraic manipulation to simplify the expression to its final form. We hope that this article has been helpful in clarifying any doubts you may have had.

Further Reading

• Algebraic Manipulation: Algebraic manipulation is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we have focused on simplifying a complex expression involving exponents and variables.
• Exponents and Variables: Exponents and variables are fundamental concepts in mathematics, and they require a deep understanding of the underlying concepts. In this article, we have focused on simplifying a complex expression involving exponents and variables.
• Mathematics: Mathematics is a vast and fascinating field that encompasses a wide range of topics, including algebra, geometry, trigonometry, and calculus. In this article, we have focused on simplifying a complex expression involving exponents and variables.

Related Articles

• Simplifying Algebraic Expressions: Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we have focused on simplifying a complex expression involving exponents and variables.
• Exponents and Variables: Exponents and variables are fundamental concepts in mathematics, and they require a deep understanding of the underlying concepts. In this article, we have focused on simplifying a complex expression involving exponents and variables.
• Mathematics: Mathematics is a vast and fascinating field that encompasses a wide range of topics, including algebra, geometry, trigonometry, and calculus. In this article, we have focused on simplifying a complex expression involving exponents and variables.