Evaluate The Expression When { A = 1$}$, { B = 2$}$, { C = 3$}$, { D = 4$}$, { E = 5$}$, And { F = 6$} . . . { \left(a^6 B^7 C^8 D E^9 F^{10}\right)^0 \}

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Understanding the Problem

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In this problem, we are given a mathematical expression and asked to evaluate it when specific values are assigned to the variables $a$, $b$, $c$, $d$, $e$, and $f$. The expression is $\left(a^6 b^7 c^8 d e^9 f^{10}\right)^0$. We need to substitute the given values of $a$, $b$, $c$, $d$, $e$, and $f$ into the expression and simplify it to find the final result.

Given Values

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The given values are:

• $a = 1$
• $b = 2$
• $c = 3$
• $d = 4$
• $e = 5$
• $f = 6$

Evaluating the Expression

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To evaluate the expression, we need to substitute the given values of $a$, $b$, $c$, $d$, $e$, and $f$ into the expression and simplify it.

$\left(a^6 b^7 c^8 d e^9 f^{10}\right)^0$

Substituting the given values:

$\left(1^6 2^7 3^8 4 5^9 6^{10}\right)^0$

Now, let's simplify the expression:

$\left(1^6 2^7 3^8 4 5^9 6^{10}\right)^0$

$= 1^0 \cdot 2^0 \cdot 3^0 \cdot 4^0 \cdot 5^0 \cdot 6^0$

$= 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1$

$= 1$

Conclusion

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In this problem, we were given a mathematical expression and asked to evaluate it when specific values were assigned to the variables $a$, $b$, $c$, $d$, $e$, and $f$. We substituted the given values into the expression and simplified it to find the final result, which is $1$.

Key Takeaways

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• When evaluating an expression with exponents, we need to follow the order of operations (PEMDAS/BODMAS).
• When raising a number to the power of $0$, the result is always $1$.
• When simplifying an expression with exponents, we can use the properties of exponents to combine like terms.

Real-World Applications

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This problem may seem simple, but it has real-world applications in various fields, such as:

• Computer Science: In computer programming, we often encounter expressions with exponents and need to evaluate them correctly.
• Engineering: In engineering, we use mathematical expressions to model and analyze complex systems, and exponents are often used to represent growth or decay rates.
• Economics: In economics, we use mathematical expressions to model economic systems and make predictions about future trends, and exponents are often used to represent growth or decay rates.

Common Mistakes

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When evaluating expressions with exponents, we often make mistakes, such as:

• Forgetting to follow the order of operations: When evaluating an expression with exponents, we need to follow the order of operations (PEMDAS/BODMAS) to ensure that we get the correct result.
• Not simplifying the expression correctly: When simplifying an expression with exponents, we need to use the properties of exponents to combine like terms and simplify the expression correctly.

Tips and Tricks

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When evaluating expressions with exponents, here are some tips and tricks to help you:

• Use the properties of exponents: When simplifying an expression with exponents, use the properties of exponents to combine like terms and simplify the expression correctly.
• Follow the order of operations: When evaluating an expression with exponents, follow the order of operations (PEMDAS/BODMAS) to ensure that you get the correct result.
• Check your work: When evaluating an expression with exponents, check your work to ensure that you have simplified the expression correctly and obtained the correct result.
  

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Frequently Asked Questions

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Q: What is the value of the expression when $a = 1$, $b = 2$, $c = 3$, $d = 4$, $e = 5$, and $f = 6$?

A: The value of the expression is $1$.

Q: Why is the value of the expression $1$?

A: The value of the expression is $1$ because when we raise any number to the power of $0$, the result is always $1$. In this case, we have $\left(a^6 b^7 c^8 d e^9 f^{10}\right)^0$, which simplifies to $1$.

Q: What is the order of operations when evaluating an expression with exponents?

A: The order of operations when evaluating an expression with exponents is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, use the properties of exponents to combine like terms. For example, if we have $a^m \cdot a^n$, we can simplify it to $a^{m+n}$.

Q: What are some common mistakes to avoid when evaluating expressions with exponents?

A: Some common mistakes to avoid when evaluating expressions with exponents include:

• Forgetting to follow the order of operations: When evaluating an expression with exponents, we need to follow the order of operations (PEMDAS/BODMAS) to ensure that we get the correct result.
• Not simplifying the expression correctly: When simplifying an expression with exponents, we need to use the properties of exponents to combine like terms and simplify the expression correctly.

Q: How do I check my work when evaluating an expression with exponents?

A: To check your work when evaluating an expression with exponents, follow these steps:

1. Re-read the problem: Make sure you understand what the problem is asking for.
2. Evaluate the expression: Evaluate the expression using the order of operations (PEMDAS/BODMAS).
3. Simplify the expression: Simplify the expression using the properties of exponents.
4. Check your work: Check your work to ensure that you have simplified the expression correctly and obtained the correct result.

Q: What are some real-world applications of evaluating expressions with exponents?

A: Some real-world applications of evaluating expressions with exponents include:

• Computer Science: In computer programming, we often encounter expressions with exponents and need to evaluate them correctly.
• Engineering: In engineering, we use mathematical expressions to model and analyze complex systems, and exponents are often used to represent growth or decay rates.
• Economics: In economics, we use mathematical expressions to model economic systems and make predictions about future trends, and exponents are often used to represent growth or decay rates.

Q: How can I practice evaluating expressions with exponents?

A: To practice evaluating expressions with exponents, try the following:

• Practice problems: Practice evaluating expressions with exponents using online resources or practice problems.
• Real-world applications: Apply the concepts of evaluating expressions with exponents to real-world problems.
• Mistake analysis: Analyze common mistakes and learn from them.

Conclusion

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Evaluating expressions with exponents is an essential skill in mathematics and has many real-world applications. By following the order of operations (PEMDAS/BODMAS) and using the properties of exponents, we can simplify expressions and obtain the correct result. Remember to check your work and practice regularly to become proficient in evaluating expressions with exponents.