Simplify The Expression: $ \left(x^2\right)^b \left(x^{3b}\right)^2 $

Introduction

In this article, we will simplify the given expression $ \left(x^2\right)b \left(x^{3b}\right)2 $. This involves applying the rules of exponents to manipulate the expression into a simpler form. We will use the properties of exponents, such as the power rule and the product rule, to simplify the expression.

Understanding the Rules of Exponents

Before we can simplify the expression, we need to understand the rules of exponents. The power rule states that for any numbers $a$ and $b$ and any integer $n$, we have:

$$\left(a^b\right)^n = a^{bn}$$

The product rule states that for any numbers $a$ and $b$ and any integer $n$, we have:

$$a^b \cdot a^c = a^{b+c}$$

Simplifying the Expression

Now that we have a good understanding of the rules of exponents, we can simplify the expression $ \left(x^2\right)b \left(x^{3b}\right)2 $. We will apply the power rule to each term in the expression.

$$\left(x^2\right)^b \left(x^{3b}\right)^2 = x^{2b} \cdot x^{6b}$$

Using the product rule, we can combine the two terms into a single term.

$$x^{2b} \cdot x^{6b} = x^{2b + 6b}$$

Simplifying the exponent, we get:

$$x^{2b + 6b} = x^{8b}$$

Conclusion

In this article, we simplified the expression $ \left(x^2\right)b \left(x^{3b}\right)2 $ using the rules of exponents. We applied the power rule and the product rule to manipulate the expression into a simpler form. The final simplified expression is $x^{8b}$.

Example Use Cases

The expression $ \left(x^2\right)b \left(x^{3b}\right)2 $ can be used in a variety of mathematical contexts. For example, it can be used to simplify expressions in algebra, calculus, and other branches of mathematics.

Step-by-Step Solution

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

1. Apply the power rule to each term in the expression.
2. Use the product rule to combine the two terms into a single term.
3. Simplify the exponent.

Common Mistakes

When simplifying expressions with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to apply the power rule or the product rule.
• Not simplifying the exponent correctly.
• Making errors when combining terms using the product rule.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions with exponents:

• Make sure to apply the power rule and the product rule correctly.
• Simplify the exponent as much as possible.
• Use the product rule to combine terms into a single term.

Conclusion

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Introduction

In our previous article, we simplified the expression $ \left(x^2\right)b \left(x^{3b}\right)2 $ using the rules of exponents. In this article, we will answer some frequently asked questions about the expression and provide additional examples and explanations.

Q&A

Q: What is the simplified form of the expression $ \left(x^2\right)b \left(x^{3b}\right)2 $?

A: The simplified form of the expression is $x^{8b}$.

Q: How do I apply the power rule to the expression?

A: To apply the power rule, you need to multiply the exponents. In this case, you would multiply $2b$ and $6b$ to get $8b$.

Q: What is the product rule, and how do I use it to simplify the expression?

A: The product rule states that for any numbers $a$ and $b$ and any integer $n$, we have:

$$a^b \cdot a^c = a^{b+c}$$

To use the product rule, you need to combine the two terms into a single term by adding the exponents.

Q: Can I use the product rule to simplify the expression $ \left(x^2\right)b \left(x^{3b}\right)2 $?

A: Yes, you can use the product rule to simplify the expression. However, you need to apply the power rule first to each term in the expression.

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

A: Some common mistakes to avoid include:

• Forgetting to apply the power rule or the product rule.
• Not simplifying the exponent correctly.
• Making errors when combining terms using the product rule.

Q: How do I simplify expressions with exponents in general?

A: To simplify expressions with exponents, you need to apply the power rule and the product rule correctly. You also need to simplify the exponent as much as possible.

Q: Can I use the expression $ \left(x^2\right)b \left(x^{3b}\right)2 $ in a real-world context?

A: Yes, you can use the expression in a real-world context. For example, you can use it to simplify expressions in algebra, calculus, and other branches of mathematics.

Example Use Cases

Here are some example use cases for the expression $ \left(x^2\right)b \left(x^{3b}\right)2 $:

• Simplifying expressions in algebra: You can use the expression to simplify expressions in algebra, such as $ \left(x^2\right)b \left(x^{3b}\right)2 = x^{8b} $.
• Simplifying expressions in calculus: You can use the expression to simplify expressions in calculus, such as $ \left(x^2\right)b \left(x^{3b}\right)2 = x^{8b} $.
• Simplifying expressions in other branches of mathematics: You can use the expression to simplify expressions in other branches of mathematics, such as number theory and combinatorics.

Step-by-Step Solution

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

1. Apply the power rule to each term in the expression.
2. Use the product rule to combine the two terms into a single term.
3. Simplify the exponent.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions with exponents:

• Forgetting to apply the power rule or the product rule.
• Not simplifying the exponent correctly.
• Making errors when combining terms using the product rule.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions with exponents:

• Make sure to apply the power rule and the product rule correctly.
• Simplify the exponent as much as possible.
• Use the product rule to combine terms into a single term.

Conclusion

In this article, we answered some frequently asked questions about the expression $ \left(x^2\right)b \left(x^{3b}\right)2 $ and provided additional examples and explanations. We also discussed some common mistakes to avoid and provided tips and tricks to help you simplify expressions with exponents.