The Expression $6 \sqrt{3^3 X^4} \cdot \sqrt{2^4 X}$ Is Equivalent To A X B A X^b A X B , Where A A A And B B B Are Positive Constants And X \textgreater 1 X \ \textgreater \ 1 X \textgreater 1 . What Is The Value Of A A A And B B B ?

Introduction

In this article, we will delve into the world of algebraic expressions and simplify a given expression to determine the values of the constants $a$ and $b$. The expression in question is $6 \sqrt{3^3 x^4} \cdot \sqrt{2^4 x}$, and we are asked to rewrite it in the form $a x^b$, where $a$ and $b$ are positive constants and $x > 1$.

Step 1: Simplifying the Expression

To simplify the given expression, we will start by using the properties of radicals. We can rewrite the expression as $6 \sqrt{3^3 x^4} \cdot \sqrt{2^4 x} = 6 \sqrt{3^3 x^4 \cdot 2^4 x}$.

Using the Properties of Radicals

We can use the property of radicals that states $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ to simplify the expression further. Applying this property, we get $6 \sqrt{3^3 x^4 \cdot 2^4 x} = 6 \sqrt{3^3 \cdot 2^4 \cdot x^5}$.

Simplifying the Radicals

We can simplify the radicals by evaluating the exponents. We have $6 \sqrt{3^3 \cdot 2^4 \cdot x^5} = 6 \sqrt{27 \cdot 16 \cdot x^5}$.

Evaluating the Exponents

We can evaluate the exponents by multiplying the numbers inside the radicals. We have $6 \sqrt{27 \cdot 16 \cdot x^5} = 6 \sqrt{432 \cdot x^5}$.

Simplifying the Radicals Again

We can simplify the radicals again by evaluating the exponents. We have $6 \sqrt{432 \cdot x^5} = 6 \sqrt{144 \cdot 3 \cdot x^5}$.

Factoring the Radicals

We can factor the radicals by taking out the perfect squares. We have $6 \sqrt{144 \cdot 3 \cdot x^5} = 6 \cdot 12 \cdot \sqrt{3 \cdot x^5}$.

Simplifying the Expression Further

We can simplify the expression further by evaluating the exponents. We have $6 \cdot 12 \cdot \sqrt{3 \cdot x^5} = 72 \cdot \sqrt{3 \cdot x^5}$.

Rewriting the Expression in the Desired Form

We can rewrite the expression in the desired form by evaluating the exponents. We have $72 \cdot \sqrt{3 \cdot x^5} = 72 \cdot \sqrt{3} \cdot \sqrt{x^5}$.

Using the Property of Radicals Again

We can use the property of radicals again to simplify the expression further. We have $72 \cdot \sqrt{3} \cdot \sqrt{x^5} = 72 \cdot \sqrt{3} \cdot x^{5/2}$.

Simplifying the Expression Again

We can simplify the expression again by evaluating the exponents. We have $72 \cdot \sqrt{3} \cdot x^{5/2} = 72 \cdot \sqrt{3} \cdot x^2 \cdot x^{1/2}$.

Using the Property of Exponents

We can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify the expression further. We have $72 \cdot \sqrt{3} \cdot x^2 \cdot x^{1/2} = 72 \cdot \sqrt{3} \cdot x^{2+1/2}$.

Evaluating the Exponents

We can evaluate the exponents by adding the numbers. We have $72 \cdot \sqrt{3} \cdot x^{2+1/2} = 72 \cdot \sqrt{3} \cdot x^{5/2}$.

Rewriting the Expression in the Desired Form Again

We can rewrite the expression in the desired form again by evaluating the exponents. We have $72 \cdot \sqrt{3} \cdot x^{5/2} = 72 \sqrt{3} x^{5/2}$.

Conclusion

In this article, we simplified the given expression $6 \sqrt{3^3 x^4} \cdot \sqrt{2^4 x}$ to determine the values of the constants $a$ and $b$. We found that the expression is equivalent to $72 \sqrt{3} x^{5/2}$, where $a = 72 \sqrt{3}$ and $b = 5/2$.

Final Answer

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Introduction

In our previous article, we simplified the expression $6 \sqrt{3^3 x^4} \cdot \sqrt{2^4 x}$ to determine the values of the constants $a$ and $b$. We found that the expression is equivalent to $72 \sqrt{3} x^{5/2}$, where $a = 72 \sqrt{3}$ and $b = 5/2$. In this article, we will answer some frequently asked questions about the simplification process and the values of $a$ and $b$.

Q: What is the first step in simplifying the expression?

A: The first step in simplifying the expression is to use the properties of radicals. We can rewrite the expression as $6 \sqrt{3^3 x^4} \cdot \sqrt{2^4 x} = 6 \sqrt{3^3 x^4 \cdot 2^4 x}$.

Q: How do we simplify the radicals in the expression?

A: We can simplify the radicals by evaluating the exponents. We have $6 \sqrt{3^3 x^4 \cdot 2^4 x} = 6 \sqrt{3^3 \cdot 2^4 \cdot x^5}$.

Q: What is the next step in simplifying the expression?

A: The next step in simplifying the expression is to simplify the radicals further by evaluating the exponents. We have $6 \sqrt{3^3 \cdot 2^4 \cdot x^5} = 6 \sqrt{27 \cdot 16 \cdot x^5}$.

Q: How do we simplify the expression further?

A: We can simplify the expression further by evaluating the exponents and factoring the radicals. We have $6 \sqrt{27 \cdot 16 \cdot x^5} = 6 \sqrt{432 \cdot x^5}$.

Q: What is the value of $a$ and $b$ in the simplified expression?

A: The value of $a$ is $72 \sqrt{3}$ and the value of $b$ is $5/2$.

Q: Why is it important to simplify the expression?

A: Simplifying the expression is important because it allows us to rewrite the expression in a more convenient form, which can make it easier to work with and understand.

Q: Can you provide an example of how to use the simplified expression?

A: Yes, here is an example of how to use the simplified expression: $72 \sqrt{3} x^{5/2} = 72 \sqrt{3} x^2 \cdot x^{1/2}$.

Q: How do we evaluate the exponents in the simplified expression?

A: We can evaluate the exponents by adding the numbers. We have $72 \sqrt{3} x^2 \cdot x^{1/2} = 72 \sqrt{3} x^{2+1/2}$.

Q: What is the final answer for the value of $a$ and $b$?

A: The final answer for the value of $a$ and $b$ is $\boxed{72 \sqrt{3}, 5/2}$.

Conclusion

In this article, we answered some frequently asked questions about the simplification process and the values of $a$ and $b$. We hope that this article has been helpful in understanding the simplification process and the values of $a$ and $b$. If you have any further questions, please don't hesitate to ask.

Final Answer

The final answer is $\boxed{72 \sqrt{3}, 5/2}$.