Generate An Equivalent Expression: ( ( 3 7 ) 4 ) 3 ⋅ ( 3 7 ) 2 \left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2 ( ( 7 3 ​ ) 4 ) 3 ⋅ ( 7 3 ​ ) 2 A. 7 3 \frac{7}{3} 3 7 ​ B. 9 49 \frac{9}{49} 49 9 ​ C. 9 7 \frac{9}{7} 7 9 ​ D. 3 7 \frac{3}{7} 7 3 ​

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

Introduction

---

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying a specific type of exponential expression, namely, the equivalent expression of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$. We will break down the problem step by step, using the properties of exponents to simplify the expression.

Understanding Exponents

---

Before we dive into the problem, let's quickly review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number. It tells us how many times to multiply the base number by itself. For example, in the expression $2^3$, the exponent $3$ tells us to multiply $2$ by itself three times: $2^3 = 2 \cdot 2 \cdot 2 = 8$.

Simplifying the Expression

---

Now that we have a basic understanding of exponents, let's tackle the problem at hand. We are given the expression $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$. Our goal is to simplify this expression.

To simplify the expression, we can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$. This property allows us to combine the exponents of the same base.

Step 1: Simplify the Inner Exponent

---

Let's start by simplifying the inner exponent, $\left(\frac{3}{7}\right)^4$. Using the property of exponents, we can rewrite this as $\left(\frac{3}{7}\right)^4 = \frac{3^4}{7^4}$.

Step 2: Simplify the Outer Exponent

---

Next, we need to simplify the outer exponent, $\left(\frac{3}{7}\right)^3$. Again, using the property of exponents, we can rewrite this as $\left(\frac{3}{7}\right)^3 = \frac{3^3}{7^3}$.

Step 3: Multiply the Exponents

---

Now that we have simplified the inner and outer exponents, we can multiply them together. Using the property of exponents that states $a^m \cdot a^n = a^{m + n}$, we can rewrite the expression as $\frac{3^4}{7^4} \cdot \frac{3^3}{7^3} = \frac{3^{4+3}}{7^{4+3}} = \frac{3^7}{7^7}$.

Step 4: Simplify the Final Expression

---

Finally, we need to simplify the final expression, $\frac{3^7}{7^7} \cdot \left(\frac{3}{7}\right)^2$. Using the property of exponents that states $a^m \cdot a^n = a^{m + n}$, we can rewrite this as $\frac{3^7}{7^7} \cdot \frac{3^2}{7^2} = \frac{3^{7+2}}{7^{7+2}} = \frac{3^9}{7^9}$.

Conclusion

---

In conclusion, we have successfully simplified the expression $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$ using the properties of exponents. The final simplified expression is $\frac{3^9}{7^9}$.

Answer

---

So, what is the equivalent expression of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$? The answer is $\boxed{\frac{3^9}{7^9}}$.

Simplifying the Answer

---

But wait, we can simplify the answer even further. Using the property of exponents that states $a^m = a^{m \cdot n}$, we can rewrite the answer as $\frac{3^9}{7^9} = \frac{3^{9 \cdot 1}}{7^{9 \cdot 1}} = \frac{3^9}{7^9} = \frac{19683}{43046721}$.

Final Answer

---

So, the final answer is $\boxed{\frac{19683}{43046721}}$. But we can simplify it even further by dividing both the numerator and the denominator by their greatest common divisor, which is $1$. Therefore, the final answer is $\boxed{\frac{19683}{43046721}}$.

Alternative Answer

---

But wait, there's an alternative answer. We can simplify the answer even further by using the property of exponents that states $a^m = a^{m \cdot n}$. This property allows us to rewrite the answer as $\frac{3^9}{79} = \frac{3^{9 \cdot 1}}{7^{9 \cdot 1}} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{79} = \frac{3^9}{

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

Q&A: Simplifying Exponential Expressions

---

Q: What is the equivalent expression of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$?

A: The equivalent expression of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$ is $\frac{3^9}{7^9}$.

Q: How do I simplify the expression $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$?

A: To simplify the expression, you can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$. This property allows you to combine the exponents of the same base.

Q: What is the property of exponents that states $(a^m)^n = a^{m \cdot n}$?

A: The property of exponents that states $(a^m)^n = a^{m \cdot n}$ is a fundamental concept in mathematics that allows you to simplify exponential expressions.

Q: How do I use the property of exponents to simplify the expression $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$?

A: To use the property of exponents to simplify the expression, you can start by simplifying the inner exponent, $\left(\frac{3}{7}\right)^4$. Using the property of exponents, you can rewrite this as $\left(\frac{3}{7}\right)^4 = \frac{3^4}{7^4}$.

Q: How do I simplify the outer exponent, $\left(\frac{3}{7}\right)^3$?

A: To simplify the outer exponent, $\left(\frac{3}{7}\right)^3$, you can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$. This property allows you to combine the exponents of the same base.

Q: What is the final simplified expression of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$?

A: The final simplified expression of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$ is $\frac{3^9}{7^9}$.

Q: Can I simplify the expression $\frac{3^9}{7^9}$ further?

A: Yes, you can simplify the expression $\frac{3^9}{7^9}$ further by using the property of exponents that states $a^m = a^{m \cdot n}$. This property allows you to rewrite the expression as $\frac{3^{9 \cdot 1}}{7^{9 \cdot 1}} = \frac{3^9}{7^9}$.

Q: What is the final answer of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$?

A: The final answer of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$ is $\boxed{\frac{3^9}{7^9}}$.

Q: Can I simplify the answer $\frac{3^9}{7^9}$ further?

A: Yes, you can simplify the answer $\frac{3^9}{7^9}$ further by dividing both the numerator and the denominator by their greatest common divisor, which is $1$. Therefore, the final answer is $\boxed{\frac{19683}{43046721}}$.

Q: What is the alternative answer of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$?

A: The alternative answer of $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$ is $\boxed{\frac{19683}{43046721}}$.

Q: Can I use the property of exponents to simplify the expression $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$?

A: Yes, you can use the property of exponents to simplify the expression $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$. This property allows you to combine the exponents of the same base.

Q: How do I use the property of exponents to simplify the expression $\left(\left(\frac{3}{7}\right)^4\right)^3 \cdot \left(\frac{3}{7}\right)^2$?

A: To use the property of exponents to simplify the expression, you can start by simplifying the inner exponent, $\left(\frac{3}{7}\right)^4$. Using the property of exponents, you can rewrite this as $\left(\frac{3}{7}\right)^4 = \frac{3^4}{7^4}$.

Q: Can I simplify the expression $\frac{3^4}{7^4} \cdot \frac{3^3}{7^3}$ further?

A: Yes, you can simplify the expression $\frac{3^4}{7^4} \cdot \frac{3^3}{7^3}$ further by using the property of exponents that states $a^m \cdot a^n = a^{m + n}$. This property allows you to combine the exponents of the same base.

Q: What is the final simplified expression of $\frac{3^4}{7^4} \cdot \frac{3^3}{7^3}$?

A: The final simplified expression of $\frac{3^4}{7^4} \cdot \frac{3^3}{7^3}$ is $\frac{3^{4+3}}{7^{4+3}} = \frac{3^7}{7^7}$.

Q: Can I simplify the expression $\frac{3^7}{7^7} \cdot \left(\frac{3}{7}\right)^2$ further?

A: Yes, you can simplify the expression $\frac{3^7}{7^7} \cdot \left(\frac{3}{7}\right)^2$ further by using the property of exponents that states $a^m \cdot a^n = a^{m + n}$. This property allows you to combine the exponents of the same base.

Q: What is the final simplified expression of $\frac{3^7}{7^7} \cdot \left(\frac{3}{7}\right)^2$?

A: The final simplified expression of $\frac{3^7}{7^7} \cdot \left(\frac{3}{7}\right)^2$ is $\frac{3^{7+2}}{7^{7+2}} = \frac{3^9}{7^9}$.

Q: Can I simplify the expression $\frac{3^9}{7^9}$ further?

A: Yes, you can simplify the expression $\frac{3^9}{7^9}$ further by dividing both the numerator and the denominator by their greatest common divisor, which is $1$. Therefore, the final answer is $\boxed{\frac{19683}{43046721}}$.

Q: What is the alternative answer of $\frac{3^9}{7^9}$?

A: The alternative answer of $\frac{3^9}{7^9}$ is $\boxed{\frac{19683}{43046721}}$.

Q: Can I use the property of exponents to simplify the expression $\frac{3^9}{7^9}$?

A: Yes, you can use the property of exponents to simplify the expression $\frac{3^9}{7^9}$. This property allows you to combine the exponents of the same base.

Q: How do I use the property of exponents to simplify the expression $\frac{3^9}{7^9}$?

A: To use the property of exponents to simplify the expression, you can start by simplifying the inner exponent, $\frac{3^9}{7^9}$. Using the property of exponents, you can rewrite this as $\frac{3^9}{7^9} = \frac{3^{9 \cdot 1}}{7^{9 \cdot 1}} = \frac{3^9}{7^9}$.

Q: Can I simplify the expression $\frac{3^9}{7^9}$ further?

A: Yes, you can simplify the expression $\frac{3^9}{7^9}$ further by dividing both the numerator and the denominator by their greatest common divisor, which is $1$.