Use The Product Rule To Simplify The Expression. Write The Result Using Exponents. \left(b^7 C\right)\left(b^{12} C^3\right ] ( B 7 C ) ( B 12 C 3 ) = □ \left(b^7 C\right)\left(b^{12} C^3\right) = \square ( B 7 C ) ( B 12 C 3 ) = □

Mar 8, 2025 by ADMIN 232 views

**Simplifying Algebraic Expressions Using the Product Rule**

Understanding the Product Rule

The product rule is a fundamental concept in algebra that allows us to simplify complex expressions by combining like terms. In this article, we will focus on using the product rule to simplify an expression involving exponents. The product rule states that when we multiply two or more variables or expressions with the same base, we can add their exponents.

The Expression to Simplify

The given expression is $\left(b^7 c\right)\left(b^{12} c^3\right)$ . Our goal is to simplify this expression using the product rule.

Applying the Product Rule

To simplify the expression, we need to apply the product rule by adding the exponents of the variables with the same base. In this case, we have two variables: $b$ and $c$ . The variable $b$ has an exponent of $7$ in the first expression and an exponent of $12$ in the second expression. Similarly, the variable $c$ has an exponent of $1$ in the first expression and an exponent of $3$ in the second expression.

Simplifying the Expression

Using the product rule, we can simplify the expression as follows:

$\left(b^7 c\right)\left(b^{12} c^3\right) = b^{7+12} c^{1+3}$

Evaluating the Exponents

Now, we need to evaluate the exponents by adding them together.

$b^{7+12} = b^{19}$

$c^{1+3} = c^4$

The Simplified Expression

Therefore, the simplified expression is:

$b^{19} c^4$

Conclusion

In this article, we used the product rule to simplify an expression involving exponents. By applying the product rule, we were able to combine like terms and simplify the expression. This is a powerful tool in algebra that allows us to simplify complex expressions and solve problems more efficiently.

Real-World Applications

The product rule has many real-world applications in fields such as physics, engineering, and computer science. For example, in physics, the product rule is used to describe the behavior of complex systems, such as the motion of objects under the influence of multiple forces. In engineering, the product rule is used to design and optimize complex systems, such as electronic circuits and mechanical systems. In computer science, the product rule is used to develop efficient algorithms for solving complex problems.

Tips and Tricks

Here are some tips and tricks for using the product rule:

Make sure to identify like terms: Before applying the product rule, make sure to identify like terms in the expression. Like terms are terms that have the same variable and exponent.
Add exponents carefully: When adding exponents, make sure to add the exponents of the variables with the same base.
Simplify the expression: After applying the product rule, simplify the expression by combining like terms.

Common Mistakes

Here are some common mistakes to avoid when using the product rule:

Not identifying like terms: Failing to identify like terms can lead to incorrect results.
Not adding exponents carefully: Failing to add exponents carefully can lead to incorrect results.
Not simplifying the expression: Failing to simplify the expression can lead to incorrect results.

Practice Problems

Here are some practice problems to help you master the product rule:

Problem 1: Simplify the expression $\left(a^3 b\right)\left(a^{12} b^2\right)$ using the product rule.
Problem 2: Simplify the expression $\left(c^4 d\right)\left(c^{10} d^3\right)$ using the product rule.
Problem 3: Simplify the expression $\left(e^2 f\right)\left(e^{15} f^4\right)$ using the product rule.

Conclusion

In conclusion, the product rule is a powerful tool in algebra that allows us to simplify complex expressions and solve problems more efficiently. By applying the product rule, we can combine like terms and simplify expressions involving exponents. With practice and patience, you can master the product rule and become proficient in simplifying algebraic expressions.

Understanding the Product Rule

The product rule is a fundamental concept in algebra that allows us to simplify complex expressions by combining like terms. In this article, we will answer some frequently asked questions about the product rule.

Q: What is the product rule?

A: The product rule is a mathematical concept that allows us to simplify complex expressions by combining like terms. It states that when we multiply two or more variables or expressions with the same base, we can add their exponents.

Q: How do I apply the product rule?

A: To apply the product rule, you need to identify like terms in the expression and add their exponents. For example, if you have the expression $\left(b^7 c\right)\left(b^{12} c^3\right)$ , you can apply the product rule by adding the exponents of the variables with the same base.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, in the expression $\left(b^7 c\right)\left(b^{12} c^3\right)$ , the terms $b^7$ and $b^{12}$ are like terms because they have the same variable $b$ .

Q: How do I simplify an expression using the product rule?

A: To simplify an expression using the product rule, you need to follow these steps:

Identify like terms in the expression.
Add the exponents of the like terms.
Simplify the expression by combining like terms.

Q: What are some common mistakes to avoid when using the product rule?

A: Some common mistakes to avoid when using the product rule include:

Not identifying like terms
Not adding exponents carefully
Not simplifying the expression

Q: How do I know if an expression can be simplified using the product rule?

A: An expression can be simplified using the product rule if it contains like terms that can be combined by adding their exponents.

Q: Can the product rule be used with negative exponents?

A: Yes, the product rule can be used with negative exponents. When adding negative exponents, you need to follow the rule that $a^{-n} = \frac{1}{a^n}$ .

Q: Can the product rule be used with fractional exponents?

A: Yes, the product rule can be used with fractional exponents. When adding fractional exponents, you need to follow the rule that $a^{m/n} = \sqrt[n]{a^m}$ .

Q: How do I apply the product rule to expressions with multiple variables?

A: To apply the product rule to expressions with multiple variables, you need to identify like terms and add their exponents. For example, if you have the expression $\left(b^7 c\right)\left(b^{12} c^3\right)\left(b^{15} c^2\right)$ , you can apply the product rule by adding the exponents of the variables with the same base.

Q: Can the product rule be used to simplify expressions with variables in the denominator?

A: Yes, the product rule can be used to simplify expressions with variables in the denominator. When simplifying expressions with variables in the denominator, you need to follow the rule that $\frac{a^m}{a^n} = a^{m-n}$ .

Conclusion

In conclusion, the product rule is a powerful tool in algebra that allows us to simplify complex expressions by combining like terms. By understanding the product rule and its applications, you can become proficient in simplifying algebraic expressions and solving problems more efficiently.

Practice Problems

Here are some practice problems to help you master the product rule:

Problem 1: Simplify the expression $\left(a^3 b\right)\left(a^{12} b^2\right)$ using the product rule.
Problem 2: Simplify the expression $\left(c^4 d\right)\left(c^{10} d^3\right)$ using the product rule.
Problem 3: Simplify the expression $\left(e^2 f\right)\left(e^{15} f^4\right)$ using the product rule.

Additional Resources

For more information on the product rule and its applications, you can refer to the following resources:

Algebra textbooks: Many algebra textbooks include a chapter on the product rule and its applications.
Online resources: There are many online resources available that provide tutorials and examples on the product rule.
Mathematical software: Many mathematical software packages, such as Mathematica and Maple, include functions for simplifying expressions using the product rule.