Simplify The Expression:$\[ 3\left(2 X^6 Y^7\right)^4 \\]

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Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression $3\left(2 x^6 y^7\right)^4$ and asked to simplify it. To do this, we need to apply the rules of exponents, specifically the power rule, which states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$.

Applying the Power Rule

Using the power rule, we can rewrite the expression as $3 \cdot 2^4 \cdot (x^6)^4 \cdot (y^7)^4$. This is because the exponent $4$ is being applied to the entire expression inside the parentheses, including the coefficients $3$ and $2$.

Simplifying the Coefficients

Now, let's simplify the coefficients. We have $3 \cdot 2^4$, which is equal to $3 \cdot 16 = 48$. So, the expression becomes $48 \cdot (x^6)^4 \cdot (y^7)^4$.

Applying the Power Rule to the Variables

Next, we need to apply the power rule to the variables $x$ and $y$. We have $(x^6)^4$ and $(y^7)^4$, which can be rewritten as $x^{6 \cdot 4}$ and $y^{7 \cdot 4}$, respectively. This gives us $x^{24}$ and $y^{28}$.

Combining the Results

Now, let's combine the results. We have $48 \cdot x^{24} \cdot y^{28}$, which is the simplified expression.

Final Answer

The final answer is $48x^{24}y^{28}$.

Understanding the Rules of Exponents

The rules of exponents are a set of rules that govern the behavior of exponents. There are several rules, including the power rule, the product rule, and the quotient rule. Understanding these rules is essential for simplifying expressions and solving equations.

The Power Rule

The power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$. This rule can be applied to both positive and negative exponents.

The Product Rule

The product rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$. This rule can be applied to both positive and negative exponents.

The Quotient Rule

The quotient rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$. This rule can be applied to both positive and negative exponents.

Examples of Simplifying Expressions

Here are some examples of simplifying expressions using the rules of exponents:

• $(2x^3y^4)^2 = 4x^6y^8$
• $(3x^2y^3)^4 = 81x^8y^{12}$
• $(4x^5y^2)^3 = 64x^{15}y^6$

Conclusion

Simplifying expressions using the rules of exponents is an essential skill for solving equations and manipulating algebraic expressions. By understanding the power rule, the product rule, and the quotient rule, you can simplify complex expressions and solve problems with ease.

Frequently Asked Questions

• Q: What is the power rule? A: The power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$.
• Q: What is the product rule? A: The product rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$.
• Q: What is the quotient rule? A: The quotient rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman

Further Reading

• [1] "Simplifying Expressions" by Khan Academy
• [2] "Rules of Exponents" by Mathway
• [3] "Algebraic Manipulation" by Wolfram MathWorld
  

$$

Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression $3\left(2 x^6 y^7\right)^4$ and asked to simplify it. To do this, we need to apply the rules of exponents, specifically the power rule, which states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$.

Q&A

Q: What is the power rule?

A: The power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$. This rule can be applied to both positive and negative exponents.

Q: How do I apply the power rule to the expression $3\left(2 x^6 y^7\right)^4$?

A: To apply the power rule, we need to rewrite the expression as $3 \cdot 2^4 \cdot (x^6)^4 \cdot (y^7)^4$. This is because the exponent $4$ is being applied to the entire expression inside the parentheses, including the coefficients $3$ and $2$.

Q: What happens to the coefficients when we apply the power rule?

A: When we apply the power rule, the coefficients are raised to the power of the exponent. In this case, we have $3 \cdot 2^4$, which is equal to $3 \cdot 16 = 48$. So, the expression becomes $48 \cdot (x^6)^4 \cdot (y^7)^4$.

Q: How do I simplify the variables $x$ and $y$?

A: To simplify the variables $x$ and $y$, we need to apply the power rule to each variable separately. We have $(x^6)^4$ and $(y^7)^4$, which can be rewritten as $x^{6 \cdot 4}$ and $y^{7 \cdot 4}$, respectively. This gives us $x^{24}$ and $y^{28}$.

Q: What is the final simplified expression?

A: The final simplified expression is $48x^{24}y^{28}$.

Q: Can you provide more examples of simplifying expressions using the power rule?

A: Here are some examples of simplifying expressions using the power rule:

• $(2x^3y^4)^2 = 4x^6y^8$
• $(3x^2y^3)^4 = 81x^8y^{12}$
• $(4x^5y^2)^3 = 64x^{15}y^6$

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

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

• Forgetting to raise the coefficients to the power of the exponent
• Forgetting to apply the power rule to each variable separately
• Not simplifying the expression correctly after applying the power rule

Q: How can I practice applying the power rule?

A: You can practice applying the power rule by working through examples and exercises in your textbook or online resources. You can also try creating your own examples and simplifying them using the power rule.

Conclusion

Simplifying expressions using the power rule is an essential skill for solving equations and manipulating algebraic expressions. By understanding the power rule and practicing its application, you can simplify complex expressions and solve problems with ease.

Frequently Asked Questions

• Q: What is the power rule? A: The power rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(ab)^m = a^mb^m$.
• Q: How do I apply the power rule to the expression $3\left(2 x^6 y^7\right)^4$? A: To apply the power rule, we need to rewrite the expression as $3 \cdot 2^4 \cdot (x^6)^4 \cdot (y^7)^4$.
• Q: What happens to the coefficients when we apply the power rule? A: When we apply the power rule, the coefficients are raised to the power of the exponent.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman

Further Reading

• [1] "Simplifying Expressions" by Khan Academy
• [2] "Rules of Exponents" by Mathway
• [3] "Algebraic Manipulation" by Wolfram MathWorld