Find The Product:$\left(\frac{1}{8} A^2 P^2\right)^2\left(7 A^3 P^4\right)^2$ = (Note: There Was An Unrelated Phrase Question Help: Vide Which Was Removed As It Is Not Part Of The Question.)

Introduction

In algebra, simplifying complex expressions is a crucial skill that helps us solve equations and inequalities more efficiently. In this article, we will focus on simplifying a specific product of two expressions, which involves exponent rules and algebraic manipulation. We will break down the problem into manageable steps and provide a clear explanation of each step.

The Problem

The given problem is to simplify the expression:

$\left(\frac{1}{8} a^2 p^2\right)^2\left(7 a^3 p^4\right)^2$

Step 1: Apply the Power Rule for Exponents

The power rule for exponents states that for any variables $x$ and $y$ and any integers $m$ and $n$, we have:

$(x^m)^n = x^{m \cdot n}$

Using this rule, we can simplify the first part of the expression:

$\left(\frac{1}{8} a^2 p^2\right)^2 = \left(\frac{1}{8}\right)^2 (a^2)^2 (p^2)^2$

Step 2: Simplify the Coefficients and Variables

Now, we can simplify the coefficients and variables separately:

$\left(\frac{1}{8}\right)^2 = \frac{1}{64}$

$(a^2)^2 = a^4$

$(p^2)^2 = p^4$

So, the first part of the expression becomes:

$\frac{1}{64} a^4 p^4$

Step 3: Apply the Power Rule for Exponents Again

Now, we can apply the power rule for exponents again to the second part of the expression:

$\left(7 a^3 p^4\right)^2 = 7^2 (a^3)^2 (p^4)^2$

Step 4: Simplify the Coefficients and Variables Again

Now, we can simplify the coefficients and variables again:

$7^2 = 49$

$(a^3)^2 = a^6$

$(p^4)^2 = p^8$

So, the second part of the expression becomes:

$49 a^6 p^8$

Step 5: Multiply the Two Simplified Expressions

Now, we can multiply the two simplified expressions:

$\frac{1}{64} a^4 p^4 \cdot 49 a^6 p^8$

Step 6: Combine Like Terms

Now, we can combine like terms:

$\frac{1}{64} \cdot 49 a^4 p^4 a^6 p^8 = \frac{49}{64} a^{10} p^{12}$

Conclusion

In this article, we simplified a complex algebraic expression by applying the power rule for exponents and combining like terms. We broke down the problem into manageable steps and provided a clear explanation of each step. By following these steps, we can simplify complex expressions and solve equations and inequalities more efficiently.

Final Answer

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Introduction

In our previous article, we simplified a complex algebraic expression by applying the power rule for exponents and combining like terms. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used in simplifying complex expressions.

Q: What is the power rule for exponents?

A: The power rule for exponents states that for any variables $x$ and $y$ and any integers $m$ and $n$, we have:

$(x^m)^n = x^{m \cdot n}$

Q: How do I apply the power rule for exponents?

A: To apply the power rule for exponents, you need to multiply the exponents of the variables. For example, if you have $(x^2)^3$, you would multiply the exponents to get $x^{2 \cdot 3} = x^6$.

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value. For example, in the expression $3x$, the number 3 is the coefficient and the letter $x$ is the variable.

Q: How do I simplify coefficients and variables?

A: To simplify coefficients and variables, you need to apply the rules of arithmetic. For example, if you have $2 \cdot 3$, you would multiply the numbers to get 6. Similarly, if you have $x^2 \cdot x^3$, you would multiply the exponents to get $x^{2+3} = x^5$.

Q: What is the difference between combining like terms and simplifying coefficients and variables?

A: Combining like terms involves adding or subtracting terms that have the same variable and exponent, while simplifying coefficients and variables involves applying the rules of arithmetic to simplify the coefficients and variables.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms that have the same variable and exponent. For example, if you have $2x + 3x$, you would combine the terms to get $5x$.

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

A: Some common mistakes to avoid when simplifying complex expressions include:

• Forgetting to apply the power rule for exponents
• Not simplifying coefficients and variables correctly
• Not combining like terms correctly
• Not checking the final answer for errors

Q: How can I practice simplifying complex expressions?

A: You can practice simplifying complex expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying complex expressions on your own and then checking your answers with a calculator or online tool.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques used in simplifying complex expressions. We covered topics such as the power rule for exponents, simplifying coefficients and variables, combining like terms, and common mistakes to avoid. By following these tips and practicing simplifying complex expressions, you can become more confident and proficient in simplifying complex expressions.

Final Tips

• Always read the problem carefully and understand what is being asked
• Break down the problem into manageable steps
• Apply the power rule for exponents and simplify coefficients and variables correctly
• Combine like terms correctly
• Check the final answer for errors

By following these tips, you can simplify complex expressions with confidence and accuracy.