Find The Simplified Form Of ( 2 S 3 E 3 4 T 4 ) 3 \left(\frac{2 S^3 E^3}{4 T^4}\right)^3 ( 4 T 4 2 S 3 E 3 ​ ) 3 . Assume That No Denominator Equals Zero.A) T 1 U 8 2 \frac{t^1 U^8}{2} 2 T 1 U 8 ​ B) L 3 8 \frac{l^3}{8} 8 L 3 ​ C) 2 R ′ W 4 32 \frac{2 R^{\prime} W^4}{32} 32 2 R ′ W 4 ​ D) 1 W 6 8 \frac{1 W^6}{8} 8 1 W 6 ​

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Introduction

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In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with exponential expressions, we need to apply the rules of exponents to simplify them. In this article, we will focus on simplifying the expression $\left(\frac{2 s^3 e^3}{4 t^4}\right)^3$. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding Exponents

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Before we dive into simplifying the expression, let's review the rules of exponents. When we have an expression in the form $a^m$, where $a$ is a number and $m$ is an exponent, we can raise it to a power using the rule:

$(a^m)^n = a^{m \cdot n}$

This rule states that when we raise a power to another power, we multiply the exponents.

Simplifying the Expression

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Now that we have reviewed the rules of exponents, let's simplify the expression $\left(\frac{2 s^3 e^3}{4 t^4}\right)^3$. To simplify this expression, we need to apply the rule of exponents mentioned earlier.

$\left(\frac{2 s^3 e^3}{4 t^4}\right)^3 = \frac{(2 s^3 e^3)^3}{(4 t^4)^3}$

Using the rule of exponents, we can simplify the numerator and denominator separately.

$(2 s^3 e^3)^3 = 2^3 \cdot (s^3)^3 \cdot (e^3)^3 = 8 \cdot s^9 \cdot e^9$

$(4 t^4)^3 = 4^3 \cdot (t^4)^3 = 64 \cdot t^{12}$

Now that we have simplified the numerator and denominator, we can rewrite the expression as:

$\frac{8 \cdot s^9 \cdot e^9}{64 \cdot t^{12}}$

Canceling Out Common Factors

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Before we simplify the expression further, let's look for common factors in the numerator and denominator. We can see that both the numerator and denominator have a factor of 8.

$\frac{8 \cdot s^9 \cdot e^9}{64 \cdot t^{12}} = \frac{s^9 \cdot e^9}{8 \cdot t^{12}}$

Simplifying the Expression Further

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Now that we have canceled out the common factor of 8, we can simplify the expression further. We can rewrite the expression as:

$\frac{s^9 \cdot e^9}{8 \cdot t^{12}} = \frac{s^9 \cdot e^9}{2^3 \cdot t^{12}}$

Applying the Rule of Exponents Again

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We can apply the rule of exponents again to simplify the expression further.

$\frac{s^9 \cdot e^9}{2^3 \cdot t^{12}} = \frac{s^{9} \cdot e^{9}}{2^3 \cdot t^{12}} = \frac{s^{9} \cdot e^{9}}{8 \cdot t^{12}}$

Simplifying the Expression to Its Final Form

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Now that we have applied the rule of exponents again, we can simplify the expression to its final form.

$\frac{s^{9} \cdot e^{9}}{8 \cdot t^{12}} = \frac{s^9 e^9}{8t^{12}}$

Conclusion

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In this article, we have simplified the expression $\left(\frac{2 s^3 e^3}{4 t^4}\right)^3$ using the rules of exponents. We have broken down the steps involved in simplifying the expression and provided a clear explanation of each step. We have also reviewed the rules of exponents and applied them to simplify the expression.

Final Answer

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The final answer to the problem is $\boxed{\frac{s^9 e^9}{8t^{12}}}$.

Comparison with Answer Choices

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Let's compare our final answer with the answer choices provided.

A) $\frac{t^1 u^8}{2}$

B) $\frac{l^3}{8}$

C) $\frac{2 r^{\prime} w^4}{32}$

D) $\frac{1 w^6}{8}$

Our final answer, $\frac{s^9 e^9}{8t^{12}}$, does not match any of the answer choices. However, we can see that the answer choices are not in the correct format. The correct format should be $\frac{s^9 e^9}{8t^{12}}$.

Discussion

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In this article, we have simplified the expression $\left(\frac{2 s^3 e^3}{4 t^4}\right)^3$ using the rules of exponents. We have broken down the steps involved in simplifying the expression and provided a clear explanation of each step. We have also reviewed the rules of exponents and applied them to simplify the expression.

The final answer to the problem is $\boxed{\frac{s^9 e^9}{8t^{12}}}$.

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Introduction

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In our previous article, we simplified the expression $\left(\frac{2 s^3 e^3}{4 t^4}\right)^3$ using the rules of exponents. In this article, we will provide a Q&A guide to help you understand the concept of simplifying exponential expressions.

Q&A

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Q1: What is the rule of exponents?

A1: The rule of exponents states that when we raise a power to another power, we multiply the exponents. In other words, $(a^m)^n = a^{m \cdot n}$.

Q2: How do we simplify an expression with exponents?

A2: To simplify an expression with exponents, we need to apply the rule of exponents. We can rewrite the expression as a product of powers and then simplify the resulting expression.

Q3: What is the difference between a power and an exponent?

A3: A power is the result of raising a number to a certain exponent. For example, $2^3$ is a power, and the exponent is 3. An exponent is the number that is raised to a certain power.

Q4: How do we simplify an expression with multiple exponents?

A4: To simplify an expression with multiple exponents, we need to apply the rule of exponents multiple times. We can rewrite the expression as a product of powers and then simplify the resulting expression.

Q5: What is the final answer to the problem?

A5: The final answer to the problem is $\boxed{\frac{s^9 e^9}{8t^{12}}}$.

Common Mistakes

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Mistake 1: Not applying the rule of exponents

A1: One common mistake is not applying the rule of exponents when simplifying an expression. Make sure to apply the rule of exponents to simplify the expression.

Mistake 2: Not canceling out common factors

A2: Another common mistake is not canceling out common factors in the numerator and denominator. Make sure to cancel out common factors to simplify the expression.

Mistake 3: Not rewriting the expression as a product of powers

A3: A third common mistake is not rewriting the expression as a product of powers. Make sure to rewrite the expression as a product of powers to simplify the expression.

Tips and Tricks

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Tip 1: Use the rule of exponents to simplify expressions

A1: Use the rule of exponents to simplify expressions. This will help you to simplify the expression quickly and efficiently.

Tip 2: Cancel out common factors

A2: Cancel out common factors in the numerator and denominator to simplify the expression.

Tip 3: Rewrite the expression as a product of powers

A3: Rewrite the expression as a product of powers to simplify the expression.

Conclusion

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In this article, we have provided a Q&A guide to help you understand the concept of simplifying exponential expressions. We have also discussed common mistakes and provided tips and tricks to help you simplify expressions efficiently.

Final Answer

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The final answer to the problem is $\boxed{\frac{s^9 e^9}{8t^{12}}}$.

Comparison with Answer Choices

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Let's compare our final answer with the answer choices provided.

A) $\frac{t^1 u^8}{2}$

B) $\frac{l^3}{8}$

C) $\frac{2 r^{\prime} w^4}{32}$

D) $\frac{1 w^6}{8}$

Our final answer, $\frac{s^9 e^9}{8t^{12}}$, does not match any of the answer choices. However, we can see that the answer choices are not in the correct format. The correct format should be $\frac{s^9 e^9}{8t^{12}}$.

Discussion

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In this article, we have provided a Q&A guide to help you understand the concept of simplifying exponential expressions. We have also discussed common mistakes and provided tips and tricks to help you simplify expressions efficiently.

The final answer to the problem is $\boxed{\frac{s^9 e^9}{8t^{12}}}$.