Simplify: \left(w^7 C\right)\left(-7 W^5 C^3\right ]

Understanding the Problem

When simplifying expressions, it's essential to apply the rules of exponents and multiplication. In this case, we're given the expression $\left(w^7 c\right)\left(-7 w^5 c^3\right)$, and we need to simplify it by combining like terms and applying the rules of exponents.

Applying the Rules of Exponents

To simplify the expression, we'll start by applying the rules of exponents. When multiplying two terms with the same base, we add their exponents. In this case, we have $w^7$ and $w^5$, which have the same base $w$. Therefore, we can add their exponents to get $w^{7+5} = w^{12}$.

Multiplying the Coefficients

In addition to applying the rules of exponents, we also need to multiply the coefficients of the two terms. The coefficient of the first term is 1, and the coefficient of the second term is -7. Therefore, we multiply these coefficients to get $1 \times -7 = -7$.

Combining the Terms

Now that we've applied the rules of exponents and multiplied the coefficients, we can combine the terms to get the simplified expression. We have $w^{12} c^3 \times -7$, which can be written as $-7 w^{12} c^3$.

Final Answer

The final answer is $\boxed{-7 w^{12} c^3}$.

Understanding the Concept of Exponents

Exponents are a fundamental concept in mathematics, and they play a crucial role in simplifying expressions. When we have a term with an exponent, such as $w^7$, it means that we need to multiply the base $w$ by itself 7 times. For example, $w^7 = w \times w \times w \times w \times w \times w \times w$.

Applying the Rules of Exponents in Real-World Scenarios

The rules of exponents are not only useful in simplifying expressions, but they also have real-world applications. For example, in physics, we often use exponents to describe the behavior of physical systems. In chemistry, we use exponents to describe the concentration of solutions.

Conclusion

In conclusion, simplifying expressions is an essential skill in mathematics, and it requires a deep understanding of the rules of exponents and multiplication. By applying these rules, we can simplify complex expressions and arrive at a final answer. Whether you're a student or a professional, understanding the rules of exponents is crucial for success in mathematics and other fields.

Frequently Asked Questions

• Q: What is the rule for multiplying two terms with the same base? A: When multiplying two terms with the same base, we add their exponents.
• Q: What is the rule for multiplying the coefficients of two terms? A: We multiply the coefficients of two terms to get the coefficient of the resulting term.
• Q: How do we simplify an expression with exponents? A: We apply the rules of exponents and multiplication to simplify the expression.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Final Thoughts

Simplifying expressions is a crucial skill in mathematics, and it requires a deep understanding of the rules of exponents and multiplication. By applying these rules, we can simplify complex expressions and arrive at a final answer. Whether you're a student or a professional, understanding the rules of exponents is crucial for success in mathematics and other fields.

Understanding the Problem

When simplifying expressions, it's essential to apply the rules of exponents and multiplication. In this case, we're given the expression $\left(w^7 c\right)\left(-7 w^5 c^3\right)$, and we need to simplify it by combining like terms and applying the rules of exponents.

Q&A

Q: What is the rule for multiplying two terms with the same base?

A: When multiplying two terms with the same base, we add their exponents. In this case, we have $w^7$ and $w^5$, which have the same base $w$. Therefore, we can add their exponents to get $w^{7+5} = w^{12}$.

Q: What is the rule for multiplying the coefficients of two terms?

A: We multiply the coefficients of two terms to get the coefficient of the resulting term. In this case, we have 1 and -7, which are the coefficients of the two terms. Therefore, we multiply these coefficients to get $1 \times -7 = -7$.

Q: How do we simplify an expression with exponents?

A: We apply the rules of exponents and multiplication to simplify the expression. In this case, we have $\left(w^7 c\right)\left(-7 w^5 c^3\right)$, which we can simplify by applying the rules of exponents and multiplication.

Q: What is the final answer to the expression $\left(w^7 c\right)\left(-7 w^5 c^3\right)$?

A: The final answer is $\boxed{-7 w^{12} c^3}$.

Q: Can you explain the concept of exponents in more detail?

A: Exponents are a fundamental concept in mathematics, and they play a crucial role in simplifying expressions. When we have a term with an exponent, such as $w^7$, it means that we need to multiply the base $w$ by itself 7 times. For example, $w^7 = w \times w \times w \times w \times w \times w \times w$.

Q: How do exponents apply to real-world scenarios?

A: The rules of exponents are not only useful in simplifying expressions, but they also have real-world applications. For example, in physics, we often use exponents to describe the behavior of physical systems. In chemistry, we use exponents to describe the concentration of solutions.

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

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

• Not applying the rules of exponents correctly
• Not multiplying the coefficients of two terms
• Not combining like terms correctly

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Final Thoughts

Simplifying expressions is a crucial skill in mathematics, and it requires a deep understanding of the rules of exponents and multiplication. By applying these rules, we can simplify complex expressions and arrive at a final answer. Whether you're a student or a professional, understanding the rules of exponents is crucial for success in mathematics and other fields.

Frequently Asked Questions

• Q: What is the rule for multiplying two terms with the same base?
• A: When multiplying two terms with the same base, we add their exponents.
• Q: What is the rule for multiplying the coefficients of two terms?
• A: We multiply the coefficients of two terms to get the coefficient of the resulting term.
• Q: How do we simplify an expression with exponents?
• A: We apply the rules of exponents and multiplication to simplify the expression.

Conclusion

In conclusion, simplifying expressions is an essential skill in mathematics, and it requires a deep understanding of the rules of exponents and multiplication. By applying these rules, we can simplify complex expressions and arrive at a final answer. Whether you're a student or a professional, understanding the rules of exponents is crucial for success in mathematics and other fields.