Use The Product Rule To Simplify The Expression.$\[ \left(a^5 B^7\right)\left(a^{15} B\right) = \square \\]

Feb 25, 2025 by ADMIN 108 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 multiplying two or more terms together. In this article, we will explore how to use the product rule to simplify the expression $\left(a^5 b^7\right)\left(a^{15} b\right)$ . This will involve applying the rules of exponents and understanding how to combine like terms.

The Product Rule Formula

The product rule formula states that when we multiply two or more terms together, we can add the exponents of the same base. In other words, if we have two terms with the same base, say $a$ , and exponents $m$ and $n$ , then the product of the two terms is equal to $a^{m+n}$ . This can be represented mathematically as:

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

Applying the Product Rule to the Given Expression

Now that we have a solid understanding of the product rule, let's apply it to the given expression $\left(a^5 b^7\right)\left(a^{15} b\right)$ . To simplify this expression, we need to multiply the two terms together and combine like terms.

Using the product rule, we can rewrite the expression as:

\left(a^5 b^7\right)\left(a^{15} b\right) = a^{5+15} \cdot b^{7+1}

Simplifying the Expression

Now that we have applied the product rule, we can simplify the expression further by combining like terms. The expression $a^{5+15}$ can be simplified to $a^{20}$ , and the expression $b^{7+1}$ can be simplified to $b^8$ .

Therefore, the simplified expression is:

a^{20} \cdot b^8

Conclusion

Real-World Applications of the Product Rule

The product rule has numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, the product rule is used to describe the motion of objects in terms of their position, velocity, and acceleration. In engineering, the product rule is used to design and optimize complex systems, such as electrical circuits and mechanical systems. In computer science, the product rule is used to develop efficient algorithms for solving complex problems.

Common Mistakes to Avoid

When applying the product rule, there are several common mistakes to avoid. These include:

Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.
Not applying the product rule correctly: Failing to apply the product rule correctly can lead to incorrect simplifications.
Not checking the final answer: Failing to check the final answer can lead to incorrect solutions.

Tips and Tricks

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

Use the product rule formula: The product rule formula is a powerful tool for simplifying complex expressions.
Combine like terms: Combining like terms is essential for simplifying complex expressions.
Check the final answer: Checking the final answer is essential for ensuring that the solution is correct.

Practice Problems

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

Problem 1: Simplify the expression $\left(a^3 b^2\right)\left(a^{10} b\right)$ using the product rule.
Problem 2: Simplify the expression $\left(a^5 c^3\right)\left(a^{15} c\right)$ using the product rule.
Problem 3: Simplify the expression $\left(b^2 d^4\right)\left(b^{10} d\right)$ using the product rule.

Conclusion

In this article, we have used the product rule to simplify the expression $\left(a^5 b^7\right)\left(a^{15} b\right)$ . We have applied the product rule formula to combine like terms and simplify the expression. The final simplified expression is $a^{20} \cdot b^8$ . This demonstrates the power of the product rule in simplifying complex algebraic expressions. We have also discussed the real-world applications of the product rule, common mistakes to avoid, and tips and tricks for applying the product rule. Finally, we have provided practice problems to help you apply the product rule.

Understanding the Product Rule

The product rule is a fundamental concept in algebra that allows us to simplify complex expressions by multiplying two or more terms together. In this article, we will explore some frequently asked questions about the product rule and provide answers to help you better understand this concept.

Q: What is the product rule?

A: The product rule is a formula that allows us to simplify complex expressions by multiplying two or more terms together. It states that when we multiply two or more terms together, we can add the exponents of the same base.

Q: How do I apply the product rule?

A: To apply the product rule, you need to identify the terms that you want to multiply together and then add the exponents of the same base. For example, if you have the expression $\left(a^3 b^2\right)\left(a^{10} b\right)$ , you can apply the product rule by adding the exponents of the same base: $a^{3+10} \cdot b^{2+1}$ .

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

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

Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.
Not applying the product rule correctly: Failing to apply the product rule correctly can lead to incorrect simplifications.
Not checking the final answer: Failing to check the final answer can lead to incorrect solutions.

Q: How do I check my answer when applying the product rule?

A: To check your answer when applying the product rule, you need to make sure that you have combined like terms correctly and that the final answer is in the simplest form possible. You can also use a calculator or a computer program to check your answer.

Q: Can I use the product rule with negative exponents?

A: Yes, you can use the product rule with negative exponents. When you have a negative exponent, you can rewrite it as a positive exponent by flipping the base and changing the sign of the exponent. For example, if you have the expression $a^{-3} \cdot b^2$ , you can rewrite it as $\frac{1}{a^3} \cdot b^2$ .

Q: Can I use the product rule with fractions?

A: Yes, you can use the product rule with fractions. When you have a fraction, you can multiply the numerator and denominator separately and then simplify the expression. For example, if you have the expression $\frac{a^3}{b^2} \cdot \frac{a^{10}}{b}$ , you can multiply the numerator and denominator separately and then simplify the expression.

Q: Can I use the product rule with variables?

A: Yes, you can use the product rule with variables. When you have a variable, you can multiply the variable by itself and then simplify the expression. For example, if you have the expression $x^3 \cdot x^{10}$ , you can multiply the variable by itself and then simplify the expression.

Q: Can I use the product rule with exponents with different bases?

A: No, you cannot use the product rule with exponents with different bases. The product rule only applies to exponents with the same base. If you have exponents with different bases, you need to use a different rule, such as the quotient rule or the power rule.

Q: Can I use the product rule with radicals?

A: No, you cannot use the product rule with radicals. The product rule only applies to exponents, not radicals. If you have radicals, you need to use a different rule, such as the quotient rule or the power rule.

Conclusion

In this article, we have answered some frequently asked questions about the product rule and provided examples to help you better understand this concept. We have also discussed some common mistakes to avoid and tips and tricks for applying the product rule. By following these tips and tricks, you can become more confident in your ability to apply the product rule and simplify complex expressions.

Practice Problems

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

Problem 1: Simplify the expression $\left(a^3 b^2\right)\left(a^{10} b\right)$ using the product rule.
Problem 2: Simplify the expression $\left(a^5 c^3\right)\left(a^{15} c\right)$ using the product rule.
Problem 3: Simplify the expression $\left(b^2 d^4\right)\left(b^{10} d\right)$ using the product rule.

Additional Resources

If you need additional help with the product rule, here are some additional resources that you can use:

Online tutorials: There are many online tutorials that can help you learn the product rule and other algebra concepts.
Practice problems: There are many practice problems available online that can help you practice applying the product rule.
Math textbooks: There are many math textbooks available that can provide you with a comprehensive understanding of the product rule and other algebra concepts.

Conclusion

In conclusion, the product rule is a fundamental concept in algebra that allows us to simplify complex expressions by multiplying two or more terms together. By following the tips and tricks in this article, you can become more confident in your ability to apply the product rule and simplify complex expressions. Remember to practice regularly and seek help when you need it. With practice and patience, you can master the product rule and become a proficient algebra student.