Perform All The Steps To Evaluate This Expression:${ \left(\frac{\left(6 7\right)\left(3 3\right)}{\left(6 6\right)\left(3 4\right)}\right)^3 }$What Is The Value Of The Expression?A. { \frac{1}{8}$}$B. { \frac{1}{2}$}$C.

Introduction

In this article, we will evaluate the given expression and provide a step-by-step guide on how to simplify it. The expression is a complex fraction that involves exponentiation and division. We will break down the expression into smaller parts and simplify each part before combining them to get the final result.

The Expression

The given expression is:

${ \left(\frac{\left(6^7\right)\left(3^3\right)}{\left(6^6\right)\left(3^4\right)}\right)^3 }$

Step 1: Simplify the Exponents

To simplify the expression, we need to start by simplifying the exponents. We can use the rule of exponents that states:

${ a^m \cdot a^n = a^{m+n} }$

Using this rule, we can simplify the exponents in the numerator and denominator:

${ \left(6^7\right)\left(3^3\right) = 6^{7+3} \cdot 3^3 = 6^{10} \cdot 3^3 }$

${ \left(6^6\right)\left(3^4\right) = 6^{6+4} \cdot 3^4 = 6^{10} \cdot 3^4 }$

Now, we can rewrite the expression as:

${ \left(\frac{6^{10} \cdot 3^3}{6^{10} \cdot 3^4}\right)^3 }$

Step 2: Cancel Out Common Factors

We can simplify the expression further by canceling out common factors in the numerator and denominator. We can use the rule of exponents that states:

${ \frac{a^m}{a^n} = a^{m-n} }$

Using this rule, we can cancel out the common factors:

${ \frac{6^{10} \cdot 3^3}{6^{10} \cdot 3^4} = \frac{3^3}{3^4} = 3^{3-4} = 3^{-1} }$

Now, we can rewrite the expression as:

${ \left(3^{-1}\right)^3 }$

Step 3: Simplify the Exponent

We can simplify the exponent by using the rule of exponents that states:

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

Using this rule, we can simplify the exponent:

${ \left(3^{-1}\right)^3 = 3^{-1 \cdot 3} = 3^{-3} }$

Now, we can rewrite the expression as:

${ 3^{-3} }$

Step 4: Evaluate the Expression

We can evaluate the expression by using the rule of exponents that states:

${ a^{-n} = \frac{1}{a^n} }$

Using this rule, we can evaluate the expression:

${ 3^{-3} = \frac{1}{3^3} = \frac{1}{27} }$

Conclusion

In this article, we evaluated the given expression and provided a step-by-step guide on how to simplify it. We broke down the expression into smaller parts and simplified each part before combining them to get the final result. The final result is:

${ \frac{1}{27} }$

This is the value of the expression.

Answer

The value of the expression is:

${ \frac{1}{27} }$

This is option A.

Discussion

This problem involves simplifying a complex fraction that involves exponentiation and division. We used the rules of exponents to simplify the expression and get the final result. This problem requires a good understanding of the rules of exponents and how to apply them to simplify complex expressions.

Related Problems

This problem is related to the following problems:

• Simplifying complex fractions that involve exponentiation and division
• Applying the rules of exponents to simplify expressions
• Evaluating expressions that involve negative exponents

Practice Problems

Try the following practice problems to test your understanding of the material:

• Simplify the expression: ${ \left(\frac{\left(2^5\right)\left(3^2\right)}{\left(2^4\right)\left(3^3\right)}\right)^2 }$
• Simplify the expression: ${ \left(\frac{\left(4^3\right)\left(5^4\right)}{\left(4^2\right)\left(5^3\right)}\right)^3 }$
• Simplify the expression: ${ \left(\frac{\left(6^2\right)\left(7^3\right)}{\left(6^3\right)\left(7^2\right)}\right)^4 }$

Conclusion

In this article, we evaluated the given expression and provided a step-by-step guide on how to simplify it. We broke down the expression into smaller parts and simplified each part before combining them to get the final result. The final result is:

${ \frac{1}{27} }$

Q&A: Evaluating the Expression

Q: What is the value of the expression?

A: The value of the expression is ${ \frac{1}{27} }$

Q: How do I simplify the expression?

A: To simplify the expression, you need to follow these steps:

1. Simplify the exponents using the rule of exponents that states: ${ a^m \cdot a^n = a^{m+n} }$
2. Cancel out common factors in the numerator and denominator using the rule of exponents that states: ${ \frac{a^m}{a^n} = a^{m-n} }$
3. Simplify the exponent using the rule of exponents that states: ${ \left(a^m\right)^n = a^{m \cdot n} }$
4. Evaluate the expression using the rule of exponents that states: ${ a^{-n} = \frac{1}{a^n} }$

Q: What is the rule of exponents?

A: The rule of exponents states that:

• ${ a^m \cdot a^n = a^{m+n} }$
• ${ \frac{a^m}{a^n} = a^{m-n} }$
• ${ \left(a^m\right)^n = a^{m \cdot n} }$
• ${ a^{-n} = \frac{1}{a^n} }$

Q: How do I apply the rule of exponents to simplify the expression?

A: To apply the rule of exponents, you need to follow these steps:

1. Identify the exponents in the numerator and denominator.
2. Simplify the exponents using the rule of exponents.
3. Cancel out common factors in the numerator and denominator.
4. Simplify the exponent.
5. Evaluate the expression.

Q: What are some common mistakes to avoid when simplifying the expression?

A: Some common mistakes to avoid when simplifying the expression include:

• Not simplifying the exponents correctly.
• Not canceling out common factors correctly.
• Not simplifying the exponent correctly.
• Not evaluating the expression correctly.

Q: How do I practice simplifying the expression?

A: To practice simplifying the expression, you can try the following:

• Simplify the expression: ${ \left(\frac{\left(2^5\right)\left(3^2\right)}{\left(2^4\right)\left(3^3\right)}\right)^2 }$
• Simplify the expression: ${ \left(\frac{\left(4^3\right)\left(5^4\right)}{\left(4^2\right)\left(5^3\right)}\right)^3 }$
• Simplify the expression: ${ \left(\frac{\left(6^2\right)\left(7^3\right)}{\left(6^3\right)\left(7^2\right)}\right)^4 }$

Q: What are some real-world applications of simplifying the expression?

A: Some real-world applications of simplifying the expression include:

• Simplifying complex fractions in algebra and calculus.
• Simplifying expressions in physics and engineering.
• Simplifying expressions in finance and economics.

Conclusion

In this article, we provided a step-by-step guide on how to simplify the expression and answered some common questions about simplifying the expression. We also provided some practice problems and real-world applications of simplifying the expression.