Simplify:$\left(64 Y^{-6} X^{12}\right)^{-\frac{1}{3}}$

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Understanding the Problem

Simplifying Exponential Expressions is a crucial concept in mathematics, and it requires a deep understanding of the properties of exponents. In this article, we will focus on simplifying the given expression $\left(64 y^{-6} x^{12}\right)^{-\frac{1}{3}}$. This involves applying the rules of exponents, including the power rule and the product rule.

Applying the Power Rule

The power rule states that for any non-zero number $a$ and integers $m$ and $n$, $\left(a^m\right)^n = a^{mn}$. We can apply this rule to simplify the given expression.

$\left(64 y^{-6} x^{12}\right)^{-\frac{1}{3}} = \left(64\right)^{-\frac{1}{3}} \left(y^{-6}\right)^{-\frac{1}{3}} \left(x^{12}\right)^{-\frac{1}{3}}$

Simplifying the Coefficients

Simplifying the Coefficients is an essential step in simplifying the expression. We can start by simplifying the coefficient $64$.

$64 = 2^6$

Using the power rule, we can rewrite the coefficient as:

$\left(64\right)^{-\frac{1}{3}} = \left(2^6\right)^{-\frac{1}{3}} = 2^{-2}$

Simplifying the Variables

Simplifying the Variables is another crucial step in simplifying the expression. We can start by simplifying the variable $y$.

$\left(y^{-6}\right)^{-\frac{1}{3}} = y^2$

Simplifying the Variables (continued)

Simplifying the Variables (continued) involves simplifying the variable $x$.

$\left(x^{12}\right)^{-\frac{1}{3}} = x^{-4}$

Combining the Simplified Terms

Combining the Simplified Terms is the final step in simplifying the expression. We can combine the simplified terms to get the final result.

$\left(64 y^{-6} x^{12}\right)^{-\frac{1}{3}} = 2^{-2} y^2 x^{-4}$

Final Result

The final result is:

$\boxed{2^{-2} y^2 x^{-4}}$

Conclusion

Simplifying the given expression $\left(64 y^{-6} x^{12}\right)^{-\frac{1}{3}}$ involves applying the rules of exponents, including the power rule and the product rule. We simplified the coefficients, variables, and combined the simplified terms to get the final result. This article demonstrates the importance of understanding the properties of exponents and how to apply them to simplify complex expressions.

Frequently Asked Questions

• What is the power rule? The power rule states that for any non-zero number $a$ and integers $m$ and $n$, $\left(a^m\right)^n = a^{mn}$.
• How do I simplify the coefficients? To simplify the coefficients, we can rewrite the coefficient as a power of a prime number and then apply the power rule.
• How do I simplify the variables? To simplify the variables, we can apply the power rule and simplify the exponent.

Additional Resources

• Exponents and Powers: This article provides a comprehensive overview of exponents and powers, including the rules of exponents and how to apply them to simplify complex expressions.
• Simplifying Expressions: This article provides a step-by-step guide on how to simplify complex expressions, including the power rule and the product rule.
• Mathematics Tutorials: This website provides a wide range of mathematics tutorials, including exponents and powers, simplifying expressions, and more.
  

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Frequently Asked Questions

Q: What is the power rule?

A: The Power Rule states that for any non-zero number $a$ and integers $m$ and $n$, $\left(a^m\right)^n = a^{mn}$. This rule allows us to simplify complex expressions by applying the exponent to the base.

Q: How do I simplify the coefficients?

A: To Simplify the Coefficients, we can rewrite the coefficient as a power of a prime number and then apply the power rule. For example, $64 = 2^6$, so we can rewrite the coefficient as $\left(2^6\right)^{-\frac{1}{3}} = 2^{-2}$.

Q: How do I simplify the variables?

A: To Simplify the Variables, we can apply the power rule and simplify the exponent. For example, $\left(y^{-6}\right)^{-\frac{1}{3}} = y^2$.

Q: What is the product rule?

A: The Product Rule states that for any non-zero numbers $a$ and $b$ and integers $m$ and $n$, $\left(ab\right)^m = a^mb^m$. This rule allows us to simplify complex expressions by applying the exponent to each factor.

Q: How do I apply the product rule?

A: To Apply the Product Rule, we can rewrite the expression as a product of two or more factors and then apply the exponent to each factor. For example, $\left(2x\right)^3 = 2^3x^3$.

Q: What is the zero exponent rule?

A: The Zero Exponent Rule states that for any non-zero number $a$, $a^0 = 1$. This rule allows us to simplify expressions with zero exponents.

Q: How do I apply the zero exponent rule?

A: To Apply the Zero Exponent Rule, we can rewrite the expression with a zero exponent as 1. For example, $x^0 = 1$.

Q: What is the negative exponent rule?

A: The Negative Exponent Rule states that for any non-zero number $a$ and integer $n$, $a^{-n} = \frac{1}{a^n}$. This rule allows us to simplify expressions with negative exponents.

Q: How do I apply the negative exponent rule?

A: To Apply the Negative Exponent Rule, we can rewrite the expression with a negative exponent as a fraction. For example, $x^{-2} = \frac{1}{x^2}$.

Additional Resources

• Exponents and Powers: This article provides a comprehensive overview of exponents and powers, including the rules of exponents and how to apply them to simplify complex expressions.
• Simplifying Expressions: This article provides a step-by-step guide on how to simplify complex expressions, including the power rule and the product rule.
• Mathematics Tutorials: This website provides a wide range of mathematics tutorials, including exponents and powers, simplifying expressions, and more.

Common Mistakes to Avoid

• Not applying the power rule: Failing to apply the power rule can lead to incorrect simplifications.
• Not simplifying the coefficients: Failing to simplify the coefficients can lead to incorrect simplifications.
• Not applying the product rule: Failing to apply the product rule can lead to incorrect simplifications.
• Not applying the zero exponent rule: Failing to apply the zero exponent rule can lead to incorrect simplifications.
• Not applying the negative exponent rule: Failing to apply the negative exponent rule can lead to incorrect simplifications.

Conclusion

Simplifying complex expressions requires a deep understanding of the rules of exponents, including the power rule, the product rule, the zero exponent rule, and the negative exponent rule. By applying these rules and avoiding common mistakes, we can simplify complex expressions and arrive at the correct solution.