Simplify The Expression:${ \frac{(2d^2 E)^2}{(a D^{-3} E 2) 3} = \frac{16}{9} }$

Mar 11, 2025 by ADMIN 82 views

Simplify the Expression: A Step-by-Step Guide to Solving the Given Equation

===========================================================

Introduction

In this article, we will delve into the world of mathematics and simplify a given expression. The expression in question is $\frac{(2d^2 e)^2}{(a d^{-3} e^2)^3} = \frac{16}{9}$ . Our goal is to simplify this expression and understand the underlying mathematical concepts that make it work.

Understanding the Expression

The given expression is a fraction, where the numerator is $(2d^2 e)^2$ and the denominator is $(a d^{-3} e^2)^3$ . To simplify this expression, we need to apply the rules of exponents and algebra.

Applying the Rules of Exponents

The first step in simplifying the expression is to apply the rules of exponents. The rule states that when we raise a power to another power, we multiply the exponents. In this case, we have $(2d^2 e)^2$ , which can be simplified to $2^2 (d^2)^2 e^2$ .

(2d^2 e)^2 = 2^2 (d^2)^2 e^2

Similarly, we can simplify the denominator by applying the rule of exponents. The denominator is $(a d^{-3} e^2)^3$ , which can be simplified to $a^3 (d^{-3})^3 (e^2)^3$ .

(a d^{-3} e^2)^3 = a^3 (d^{-3})^3 (e^2)^3

Simplifying the Expression

Now that we have applied the rules of exponents, we can simplify the expression by canceling out common factors. The numerator is $2^2 (d^2)^2 e^2$ , and the denominator is $a^3 (d^{-3})^3 (e^2)^3$ . We can cancel out the common factors of $e^2$ in the numerator and denominator.

\frac{2^2 (d^2)^2 e^2}{a^3 (d^{-3})^3 (e^2)^3} = \frac{2^2 (d^2)^2}{a^3 (d^{-3})^3}

Next, we can simplify the expression by applying the rule of exponents. We have $(d^2)^2$ in the numerator, which can be simplified to $d^4$ . Similarly, we have $(d^{-3})^3$ in the denominator, which can be simplified to $d^{-9}$ .

\frac{2^2 (d^2)^2}{a^3 (d^{-3})^3} = \frac{2^2 d^4}{a^3 d^{-9}}

Canceling Out Common Factors

Now that we have simplified the expression, we can cancel out common factors. We have $d^4$ in the numerator and $d^{-9}$ in the denominator. We can cancel out the common factor of $d^4$ in the numerator and denominator.

\frac{2^2 d^4}{a^3 d^{-9}} = \frac{2^2}{a^3 d^{-9} d^4}

Next, we can simplify the expression by applying the rule of exponents. We have $d^{-9} d^4$ in the denominator, which can be simplified to $d^{-5}$ .

\frac{2^2}{a^3 d^{-9} d^4} = \frac{2^2}{a^3 d^{-5}}

Simplifying the Expression Further

Now that we have simplified the expression, we can simplify it further by applying the rule of exponents. We have $2^2$ in the numerator, which can be simplified to $4$ . Similarly, we have $a^3$ in the denominator, which can be simplified to $a^3$ .

\frac{2^2}{a^3 d^{-5}} = \frac{4}{a^3 d^{-5}}

Equating the Expression to the Given Value

Now that we have simplified the expression, we can equate it to the given value of $\frac{16}{9}$ . We can set up an equation by equating the two expressions.

\frac{4}{a^3 d^{-5}} = \frac{16}{9}

Solving for a and d

Now that we have equated the expression to the given value, we can solve for $a$ and $d$ . We can start by cross-multiplying the equation.

9 \cdot 4 = 16 \cdot a^3 d^{-5}

Next, we can simplify the equation by applying the rule of exponents. We have $9 \cdot 4$ on the left-hand side, which can be simplified to $36$ . Similarly, we have $16 \cdot a^3 d^{-5}$ on the right-hand side, which can be simplified to $16a^3d^{-5}$ .

36 = 16a^3d^{-5}

Canceling Out Common Factors

Now that we have simplified the equation, we can cancel out common factors. We have $4$ on both sides of the equation, which can be canceled out.

9 = 4a^3d^{-5}

Next, we can simplify the equation by applying the rule of exponents. We have $4a^3d^{-5}$ on the right-hand side, which can be simplified to $4a^3d^{-5}$ .

9 = 4a^3d^{-5}

Solving for a and d

Now that we have simplified the equation, we can solve for $a$ and $d$ . We can start by dividing both sides of the equation by $4$ .

\frac{9}{4} = a^3d^{-5}

Next, we can simplify the equation by applying the rule of exponents. We have $a^3d^{-5}$ on the right-hand side, which can be simplified to $a^3d^{-5}$ .

\frac{9}{4} = a^3d^{-5}

Taking the Cube Root of Both Sides

Now that we have simplified the equation, we can take the cube root of both sides. This will allow us to solve for $a$ .

\sqrt[3]{\frac{9}{4}} = \sqrt[3]{a^3d^{-5}}

Next, we can simplify the equation by applying the rule of exponents. We have $\sqrt[3]{a^3d^{-5}}$ on the right-hand side, which can be simplified to $ad^{-5/3}$ .

\sqrt[3]{\frac{9}{4}} = ad^{-5/3}

Solving for a

Now that we have simplified the equation, we can solve for $a$ . We can start by isolating $a$ on one side of the equation.

a = \sqrt[3]{\frac{9}{4}}d^{5/3}

Solving for d

Now that we have solved for $a$ , we can solve for $d$ . We can start by isolating $d$ on one side of the equation.

d = \sqrt[3]{\frac{4}{9}}a^{-5/3}

Conclusion

In this article, we simplified the expression $\frac{(2d^2 e)^2}{(a d^{-3} e^2)^3} = \frac{16}{9}$ . We applied the rules of exponents and algebra to simplify the expression and solve for $a$ and $d$ . The final solution is $a = \sqrt[3]{\frac{9}{4}}d^{5/3}$ and $d = \sqrt[3]{\frac{4}{9}}a^{-5/3}$ .

=====================================

Introduction

In our previous article, we simplified the expression $\frac{(2d^2 e)^2}{(a d^{-3} e^2)^3} = \frac{16}{9}$ . We applied the rules of exponents and algebra to simplify the expression and solve for $a$ and $d$ . In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q&A

Q: What is the final solution to the expression?

A: The final solution to the expression is $a = \sqrt[3]{\frac{9}{4}}d^{5/3}$ and $d = \sqrt[3]{\frac{4}{9}}a^{-5/3}$ .

Q: How did you simplify the expression?

A: We applied the rules of exponents and algebra to simplify the expression. We started by applying the rule of exponents to the numerator and denominator, and then we canceled out common factors.

Q: What is the rule of exponents?

A: The rule of exponents states that when we raise a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{mn}$ .

Q: How do you apply the rule of exponents?

A: To apply the rule of exponents, we simply multiply the exponents. For example, $(a^2)^3 = a^{2 \cdot 3} = a^6$ .

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

A: A power is the result of raising a number to a certain exponent. For example, $a^2$ is a power of $a$ . An exponent is the number that is raised to a certain power. For example, $2$ is an exponent in the expression $a^2$ .

Q: How do you simplify an expression with exponents?

A: To simplify an expression with exponents, we can apply the rule of exponents and cancel out common factors. We can also use the properties of exponents, such as the product rule and the quotient rule.

Q: What is the product rule?

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

Q: What is the quotient rule?

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

Q: How do you solve for a and d?

A: To solve for $a$ and $d$ , we can isolate them on one side of the equation. We can use the properties of exponents and algebra to solve for $a$ and $d$ .

Q: What is the difference between a and d?

A: $a$ and $d$ are variables that represent different values. $a$ represents the value of $a$ , and $d$ represents the value of $d$ .

Q: How do you take the cube root of a number?

A: To take the cube root of a number, we can use the formula $\sqrt[3]{x} = \sqrt[3]{x}$ . We can also use a calculator to find the cube root of a number.

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when raised to the power of 3, equals the original number. For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$ .

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the expression $\frac{(2d^2 e)^2}{(a d^{-3} e^2)^3} = \frac{16}{9}$ . We answered questions about the rule of exponents, how to simplify an expression with exponents, and how to solve for $a$ and $d$ . We hope this guide has been helpful in understanding the solution to the expression.

Additional Resources

Final Thoughts

We hope this Q&A guide has been helpful in understanding the solution to the expression $\frac{(2d^2 e)^2}{(a d^{-3} e^2)^3} = \frac{16}{9}$ . If you have any further questions or need additional help, please don't hesitate to ask.