Simplify ( 6 G 4 H 5 ) 2 \left(6 G^4 H^5\right)^2 ( 6 G 4 H 5 ) 2 .A. 12 G 6 H 7 12 G^6 H^7 12 G 6 H 7 B. 12 G 8 H 10 12 G^8 H^{10} 12 G 8 H 10 C. 36 G 6 H 7 36 G^6 H^7 36 G 6 H 7 D. 36 G 8 H 10 36 G^8 H^{10} 36 G 8 H 10

Understanding Exponents and Power Rules

In mathematics, exponents and power rules are essential concepts that help us simplify complex expressions. When dealing with exponents, we need to understand the rules that govern their behavior. In this article, we will focus on simplifying the expression $\left(6 g^4 h^5\right)^2$ using the power rule.

The Power Rule

The power rule states that for any non-zero number $a$ and integers $m$ and $n$, we have:

$$\left(a^m\right)^n = a^{m \cdot n}$$

This rule allows us to simplify expressions by multiplying the exponents when raising a power to another power.

Simplifying the Expression

Now, let's apply the power rule to simplify the expression $\left(6 g^4 h^5\right)^2$. We can start by breaking down the expression into its components:

$$\left(6 g^4 h^5\right)^2 = \left(6\right)^2 \cdot \left(g^4\right)^2 \cdot \left(h^5\right)^2$$

Using the power rule, we can simplify each component:

$$\left(6\right)^2 = 6^2 = 36$$

$$\left(g^4\right)^2 = g^{4 \cdot 2} = g^8$$

$$\left(h^5\right)^2 = h^{5 \cdot 2} = h^{10}$$

Now, we can combine the simplified components:

$$\left(6 g^4 h^5\right)^2 = 36 g^8 h^{10}$$

Conclusion

In this article, we simplified the expression $\left(6 g^4 h^5\right)^2$ using the power rule. We broke down the expression into its components, applied the power rule to each component, and then combined the simplified components to get the final result. The correct answer is:

$36 g^8 h^{10}$

This result is consistent with the power rule, which states that when raising a power to another power, we multiply the exponents.

Key Takeaways

• The power rule states that for any non-zero number $a$ and integers $m$ and $n$, we have: $\left(a^m\right)^n = a^{m \cdot n}$
• To simplify an expression using the power rule, break it down into its components, apply the power rule to each component, and then combine the simplified components.
• The power rule is a fundamental concept in mathematics that helps us simplify complex expressions.

Practice Problems

1. Simplify the expression $\left(3 x^2 y^3\right)^2$ using the power rule.
2. Simplify the expression $\left(2 a^3 b^4\right)^3$ using the power rule.
3. Simplify the expression $\left(5 c^2 d^3\right)^4$ using the power rule.

Answer Key

1. $9 x^4 y^6$
2. $8 a^9 b^{12}$
3. $625 c^8 d^{12}$
   Frequently Asked Questions (FAQs) =====================================

Q: What is the power rule in mathematics?

A: The power rule is a fundamental concept in mathematics that states that for any non-zero number $a$ and integers $m$ and $n$, we have: $\left(a^m\right)^n = a^{m \cdot n}$. This rule allows us to simplify expressions by multiplying the exponents when raising a power to another power.

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

A: To apply the power rule, break down the expression into its components, apply the power rule to each component, and then combine the simplified components. For example, to simplify the expression $\left(6 g^4 h^5\right)^2$, we can break it down into its components and apply the power rule as follows:

$$\left(6 g^4 h^5\right)^2 = \left(6\right)^2 \cdot \left(g^4\right)^2 \cdot \left(h^5\right)^2$$

Using the power rule, we can simplify each component:

$$\left(6\right)^2 = 6^2 = 36$$

$$\left(g^4\right)^2 = g^{4 \cdot 2} = g^8$$

$$\left(h^5\right)^2 = h^{5 \cdot 2} = h^{10}$$

Now, we can combine the simplified components:

$$\left(6 g^4 h^5\right)^2 = 36 g^8 h^{10}$$

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:

• Not breaking down the expression into its components
• Not applying the power rule to each component
• Not combining the simplified components correctly
• Not checking the result for errors

Q: Can I use the power rule to simplify expressions with negative exponents?

A: Yes, you can use the power rule to simplify expressions with negative exponents. For example, to simplify the expression $\left(x^{-2}\right)^3$, we can apply the power rule as follows:

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

Q: Can I use the power rule to simplify expressions with fractional exponents?

A: Yes, you can use the power rule to simplify expressions with fractional exponents. For example, to simplify the expression $\left(x^{1/2}\right)^3$, we can apply the power rule as follows:

$$\left(x^{1/2}\right)^3 = x^{1/2 \cdot 3} = x^{3/2}$$

Q: What are some real-world applications of the power rule?

A: The power rule has many real-world applications, including:

• Simplifying complex expressions in algebra and calculus
• Solving equations and inequalities
• Modeling real-world phenomena, such as population growth and chemical reactions
• Optimizing functions and solving optimization problems

Conclusion

In this article, we answered some frequently asked questions about the power rule, including how to apply it, common mistakes to avoid, and real-world applications. We hope this article has been helpful in clarifying the power rule and its applications.