Geoffrey Is Evaluating The Expression ( − 3 ) 3 ( 2 6 ) ( − 3 ) 5 ( 2 2 ) \frac{(-3)^3\left(2^6\right)}{(-3)^5\left(2^2\right)} ( − 3 ) 5 ( 2 2 ) ( − 3 ) 3 ( 2 6 ) ​ As Shown Below: ( − 3 ) 3 ( 2 6 ) ( − 3 ) 5 ( 2 2 ) = ( 2 ) A ( − 3 ) B = C D \frac{(-3)^3\left(2^6\right)}{(-3)^5\left(2^2\right)} = \frac{(2)^a}{(-3)^b} = \frac{c}{d} ( − 3 ) 5 ( 2 2 ) ( − 3 ) 3 ( 2 6 ) ​ = ( − 3 ) B ( 2 ) A ​ = D C ​ What Are The Values Of $a, B,

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Understanding the Problem

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Geoffrey is evaluating the expression $\frac{(-3)^3\left(2^6\right)}{(-3)^5\left(2^2\right)}$ as shown below:

$\frac{(-3)^3\left(2^6\right)}{(-3)^5\left(2^2\right)} = \frac{(2)^a}{(-3)^b} = \frac{c}{d}$

The goal is to simplify the given expression and find the values of $a, b, c,$ and $d$.

Breaking Down the Expression

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To simplify the expression, we need to apply the rules of exponents. The expression can be broken down into two parts: the numerator and the denominator.

Numerator

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The numerator is $(-3)^3\left(2^6\right)$. We can simplify this part by applying the rule of exponents that states when we multiply two numbers with the same base, we add their exponents.

$(-3)^3\left(2^6\right) = (-3)^3 \cdot 2^6$

Denominator

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The denominator is $(-3)^5\left(2^2\right)$. We can simplify this part by applying the same rule of exponents.

$(-3)^5\left(2^2\right) = (-3)^5 \cdot 2^2$

Simplifying the Expression

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Now that we have broken down the expression into two parts, we can simplify it by dividing the numerator by the denominator.

$\frac{(-3)^3\left(2^6\right)}{(-3)^5\left(2^2\right)} = \frac{(-3)^3 \cdot 2^6}{(-3)^5 \cdot 2^2}$

We can simplify this expression by applying the rule of exponents that states when we divide two numbers with the same base, we subtract their exponents.

$\frac{(-3)^3 \cdot 2^6}{(-3)^5 \cdot 2^2} = \frac{(-3)^{3-5} \cdot 2^{6-2}}{1}$

$\frac{(-3)^3 \cdot 2^6}{(-3)^5 \cdot 2^2} = \frac{(-3)^{-2} \cdot 2^4}{1}$

Finding the Values of $a, b, c,$ and $d$

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Now that we have simplified the expression, we can find the values of $a, b, c,$ and $d$.

$\frac{(-3)^{-2} \cdot 2^4}{1} = \frac{(2)^4}{(-3)^{-2}} = \frac{c}{d}$

We can see that $a = 4$ and $b = -2$. The value of $c$ is $2^4 = 16$ and the value of $d$ is $(-3)^{-2} = -\frac{1}{9}$.

Conclusion

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In conclusion, we have simplified the given expression and found the values of $a, b, c,$ and $d$. The expression can be simplified to $\frac{16}{-\frac{1}{9}}$ and the values of $a, b, c,$ and $d$ are $4, -2, 16,$ and $-\frac{1}{9}$ respectively.

Final Answer

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The final answer is $\boxed{4, -2, 16, -\frac{1}{9}}$.

Step-by-Step Solution

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Here is the step-by-step solution to the problem:

1. Break down the expression into two parts: the numerator and the denominator.
2. Simplify the numerator by applying the rule of exponents that states when we multiply two numbers with the same base, we add their exponents.
3. Simplify the denominator by applying the same rule of exponents.
4. Simplify the expression by dividing the numerator by the denominator.
5. Apply the rule of exponents that states when we divide two numbers with the same base, we subtract their exponents.
6. Find the values of $a, b, c,$ and $d$ by simplifying the expression.

Key Concepts

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The key concepts in this problem are:

• The rule of exponents that states when we multiply two numbers with the same base, we add their exponents.
• The rule of exponents that states when we divide two numbers with the same base, we subtract their exponents.
• Simplifying expressions by applying the rules of exponents.

Real-World Applications

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The concept of simplifying expressions by applying the rules of exponents has many real-world applications. For example, in physics, we often need to simplify complex expressions to solve problems. In engineering, we need to simplify expressions to design and build complex systems. In finance, we need to simplify expressions to calculate interest rates and investments.

Common Mistakes

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The most common mistakes in this problem are:

• Not breaking down the expression into two parts: the numerator and the denominator.
• Not applying the rules of exponents correctly.
• Not simplifying the expression by dividing the numerator by the denominator.

Tips and Tricks

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Here are some tips and tricks to help you solve this problem:

• Make sure to break down the expression into two parts: the numerator and the denominator.
• Apply the rules of exponents correctly.
• Simplify the expression by dividing the numerator by the denominator.
• Check your work by plugging in the values of $a, b, c,$ and $d$ into the original expression.

Practice Problems

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Here are some practice problems to help you practice simplifying expressions by applying the rules of exponents:

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

Conclusion

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In conclusion, we have simplified the given expression and found the values of $a, b, c,$ and $d$. The expression can be simplified to $\frac{16}{-\frac{1}{9}}$ and the values of $a, b, c,$ and $d$ are $4, -2, 16,$ and $-\frac{1}{9}$ respectively.

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Q: What is the rule of exponents for multiplying numbers with the same base?

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A: The rule of exponents for multiplying numbers with the same base is to add their exponents. For example, if we have $(2)^3 \cdot (2)^4$, we can simplify it to $(2)^{3+4} = (2)^7$.

Q: What is the rule of exponents for dividing numbers with the same base?

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A: The rule of exponents for dividing numbers with the same base is to subtract their exponents. For example, if we have $\frac{(2)^3}{(2)^4}$, we can simplify it to $(2)^{3-4} = (2)^{-1}$.

Q: How do I simplify an expression with exponents?

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A: To simplify an expression with exponents, you need to follow these steps:

1. Break down the expression into two parts: the numerator and the denominator.
2. Simplify the numerator by applying the rule of exponents for multiplying numbers with the same base.
3. Simplify the denominator by applying the rule of exponents for multiplying numbers with the same base.
4. Simplify the expression by dividing the numerator by the denominator.
5. Apply the rule of exponents for dividing numbers with the same base.

Q: What is the difference between a positive exponent and a negative exponent?

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A: A positive exponent represents a power of a number, while a negative exponent represents a reciprocal of a power of a number. For example, $(2)^3 = 8$ and $(2)^{-3} = \frac{1}{(2)^3} = \frac{1}{8}$.

Q: How do I handle exponents with different bases?

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A: When you have exponents with different bases, you need to apply the rule of exponents for multiplying numbers with the same base. For example, if we have $(2)^3 \cdot (3)^4$, we can simplify it to $(2)^3 \cdot (3)^4$.

Q: Can I simplify an expression with exponents that have different bases?

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A: Yes, you can simplify an expression with exponents that have different bases by applying the rule of exponents for multiplying numbers with the same base. For example, if we have $\frac{(2)^3}{(3)^4}$, we can simplify it to $\frac{(2)^3}{(3)^4}$.

Q: How do I handle fractions with exponents?

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A: When you have fractions with exponents, you need to apply the rule of exponents for dividing numbers with the same base. For example, if we have $\frac{(2)^3}{(2)^4}$, we can simplify it to $(2)^{3-4} = (2)^{-1}$.

Q: Can I simplify an expression with exponents that have fractions?

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A: Yes, you can simplify an expression with exponents that have fractions by applying the rule of exponents for dividing numbers with the same base. For example, if we have $\frac{(2)^3}{\frac{(2)^4}{(3)^2}}$, we can simplify it to $(2)^{3-4} \cdot (3)^2 = (2)^{-1} \cdot (3)^2$.

Q: How do I handle exponents with variables?

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A: When you have exponents with variables, you need to apply the rule of exponents for multiplying numbers with the same base. For example, if we have $(x)^3 \cdot (x)^4$, we can simplify it to $(x)^{3+4} = (x)^7$.

Q: Can I simplify an expression with exponents that have variables?

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A: Yes, you can simplify an expression with exponents that have variables by applying the rule of exponents for multiplying numbers with the same base. For example, if we have $\frac{(x)^3}{(x)^4}$, we can simplify it to $(x)^{3-4} = (x)^{-1}$.

Q: How do I handle exponents with negative bases?

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A: When you have exponents with negative bases, you need to apply the rule of exponents for multiplying numbers with the same base. For example, if we have $(-2)^3$, we can simplify it to $-8$.

Q: Can I simplify an expression with exponents that have negative bases?

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A: Yes, you can simplify an expression with exponents that have negative bases by applying the rule of exponents for multiplying numbers with the same base. For example, if we have $\frac{(-2)^3}{(-2)^4}$, we can simplify it to $(-2)^{3-4} = (-2)^{-1}$.

Q: How do I handle exponents with zero bases?

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A: When you have exponents with zero bases, you need to apply the rule of exponents for multiplying numbers with the same base. For example, if we have $(0)^3$, we can simplify it to $0$.

Q: Can I simplify an expression with exponents that have zero bases?

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A: Yes, you can simplify an expression with exponents that have zero bases by applying the rule of exponents for multiplying numbers with the same base. For example, if we have $\frac{(0)^3}{(0)^4}$, we can simplify it to $(0)^{3-4} = (0)^{-1}$.

Q: How do I handle exponents with undefined bases?

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A: When you have exponents with undefined bases, you need to apply the rule of exponents for multiplying numbers with the same base. For example, if we have $(\sqrt{-1})^3$, we can simplify it to $-i$.

Q: Can I simplify an expression with exponents that have undefined bases?

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A: Yes, you can simplify an expression with exponents that have undefined bases by applying the rule of exponents for multiplying numbers with the same base. For example, if we have $\frac{(\sqrt{-1})^3}{(\sqrt{-1})^4}$, we can simplify it to $(\sqrt{-1})^{3-4} = (\sqrt{-1})^{-1}$.

Conclusion

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In conclusion, we have discussed the rules of exponents and how to simplify expressions with exponents. We have also answered some common questions about exponents and provided examples to illustrate the concepts.