The Expression $\left(3 A^4 B^4 C^4\right)^2\left(2 A^5 B^5 C^5\right)^3$ Equals $n A^r B^s C^t$ Where:- $n$, The Leading Coefficient, Is: $\square$ 72- $r$, The Exponent Of $a$, Is:

Introduction

In algebra, the process of simplifying complex expressions is a crucial skill that helps in solving various mathematical problems. One such problem involves simplifying the given expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ to the form $n a^r b^s c^t$. This problem requires us to apply the rules of exponents and simplify the given expression step by step.

Understanding the Rules of Exponents

Before we proceed with simplifying the given expression, it is essential to understand the rules of exponents. The rules of exponents state that when we multiply two or more numbers with the same base, we add their exponents. On the other hand, when we divide two or more numbers with the same base, we subtract their exponents.

Simplifying the Given Expression

To simplify the given expression, we will first apply the rule of exponents that states $\left(a^m\right)^n = a^{mn}$. Using this rule, we can rewrite the given expression as follows:

$\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3 = \left(3^2 a^{4 \cdot 2} b^{4 \cdot 2} c^{4 \cdot 2}\right)\left(2^3 a^{5 \cdot 3} b^{5 \cdot 3} c^{5 \cdot 3}\right)$

Applying the Rule of Exponents

Now that we have rewritten the given expression, we can apply the rule of exponents that states $a^m \cdot a^n = a^{m+n}$. Using this rule, we can simplify the expression further as follows:

$\left(3^2 a^{4 \cdot 2} b^{4 \cdot 2} c^{4 \cdot 2}\right)\left(2^3 a^{5 \cdot 3} b^{5 \cdot 3} c^{5 \cdot 3}\right) = 3^2 \cdot 2^3 a^{4 \cdot 2 + 5 \cdot 3} b^{4 \cdot 2 + 5 \cdot 3} c^{4 \cdot 2 + 5 \cdot 3}$

Simplifying the Exponents

Now that we have applied the rule of exponents, we can simplify the exponents further. Using the rule that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression as follows:

$3^2 \cdot 2^3 a^{4 \cdot 2 + 5 \cdot 3} b^{4 \cdot 2 + 5 \cdot 3} c^{4 \cdot 2 + 5 \cdot 3} = 3^2 \cdot 2^3 a^{8 + 15} b^{8 + 15} c^{8 + 15}$

Evaluating the Exponents

Now that we have simplified the exponents, we can evaluate them further. Using the rule that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression as follows:

$3^2 \cdot 2^3 a^{8 + 15} b^{8 + 15} c^{8 + 15} = 3^2 \cdot 2^3 a^{23} b^{23} c^{23}$

Evaluating the Leading Coefficient

Now that we have simplified the expression, we can evaluate the leading coefficient. The leading coefficient is the product of the coefficients of the terms in the expression. In this case, the leading coefficient is $3^2 \cdot 2^3$.

Evaluating the Exponents of $a$, $b$, and $c$

Now that we have simplified the expression, we can evaluate the exponents of $a$, $b$, and $c$. The exponent of $a$ is 23, the exponent of $b$ is 23, and the exponent of $c$ is 23.

Conclusion

In conclusion, the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ equals $n a^r b^s c^t$ where $n = 3^2 \cdot 2^3 = 72$, $r = 23$, $s = 23$, and $t = 23$.

Discussion

• What is the leading coefficient of the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$?
• What are the exponents of $a$, $b$, and $c$ in the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$?
• How do you simplify the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ using the rules of exponents?

Answer

• The leading coefficient of the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ is $3^2 \cdot 2^3 = 72$.
• The exponents of $a$, $b$, and $c$ in the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ are 23, 23, and 23 respectively.
• To simplify the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ using the rules of exponents, we can apply the rule that states $\left(a^m\right)^n = a^{mn}$ and the rule that states $a^m \cdot a^n = a^{m+n}$.
  

Introduction

In our previous article, we simplified the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ to the form $n a^r b^s c^t$. In this article, we will answer some frequently asked questions related to the expression and its simplification.

Q&A

Q1: What is the leading coefficient of the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$?

A1: The leading coefficient of the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ is $3^2 \cdot 2^3 = 72$.

Q2: What are the exponents of $a$, $b$, and $c$ in the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$?

A2: The exponents of $a$, $b$, and $c$ in the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ are 23, 23, and 23 respectively.

Q3: How do you simplify the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ using the rules of exponents?

A3: To simplify the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ using the rules of exponents, we can apply the rule that states $\left(a^m\right)^n = a^{mn}$ and the rule that states $a^m \cdot a^n = a^{m+n}$.

Q4: What is the value of $n$ in the expression $n a^r b^s c^t$?

A4: The value of $n$ in the expression $n a^r b^s c^t$ is $3^2 \cdot 2^3 = 72$.

Q5: What are the values of $r$, $s$, and $t$ in the expression $n a^r b^s c^t$?

A5: The values of $r$, $s$, and $t$ in the expression $n a^r b^s c^t$ are 23, 23, and 23 respectively.

Q6: How do you evaluate the exponents of $a$, $b$, and $c$ in the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$?

A6: To evaluate the exponents of $a$, $b$, and $c$ in the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$, we can apply the rule that states $\left(a^m\right)^n = a^{mn}$ and the rule that states $a^m \cdot a^n = a^{m+n}$.

Q7: What is the final form of the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$?

A7: The final form of the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ is $72 a^{23} b^{23} c^{23}$.

Conclusion

In conclusion, we have answered some frequently asked questions related to the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ and its simplification. We hope that this article has provided valuable information and insights to our readers.

Discussion

• What is the leading coefficient of the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$?
• What are the exponents of $a$, $b$, and $c$ in the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$?
• How do you simplify the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ using the rules of exponents?

Answer

• The leading coefficient of the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ is $3^2 \cdot 2^3 = 72$.
• The exponents of $a$, $b$, and $c$ in the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ are 23, 23, and 23 respectively.
• To simplify the expression $\left(3 a^4 b^4 c^4\right)^2\left(2 a^5 b^5 c^5\right)^3$ using the rules of exponents, we can apply the rule that states $\left(a^m\right)^n = a^{mn}$ and the rule that states $a^m \cdot a^n = a^{m+n}$.