Simplify The Expression: $\left(2 Q^5\right)^4$A. $2 Q^{20}$ B. $2 Q^{625}$ C. $16 Q^{20}$ D. $16 Q^9$

Simplify the Expression: $\left(2 q^5\right)^4$

Understanding Exponents and Power Rules

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

The Power Rule of Exponents

The power rule of exponents states that for any non-zero number $a$ and integers $m$ and $n$, the following rule holds:

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

This rule allows us to simplify expressions by multiplying the exponents when the base is the same.

Applying the Power Rule to the Expression

Now, let's apply the power rule to the given expression $\left(2 q^5\right)^4$. According to the power rule, we can simplify this expression by multiplying the exponents:

$$\left(2 q^5\right)^4 = 2^4 \cdot \left(q^5\right)^4$$

Simplifying the Expression

Using the power rule, we can simplify the expression further:

$$2^4 \cdot \left(q^5\right)^4 = 16 \cdot q^{5 \cdot 4}$$

Evaluating the Exponent

Now, let's evaluate the exponent $5 \cdot 4$:

$$5 \cdot 4 = 20$$

Final Simplified Expression

Substituting the evaluated exponent back into the expression, we get:

$$16 \cdot q^{20}$$

Conclusion

In this article, we simplified the expression $\left(2 q^5\right)^4$ using the power rule of exponents. By applying the power rule, we were able to simplify the expression and arrive at the final answer: $16 q^{20}$. This example demonstrates the importance of understanding exponents and power rules in mathematics.

Comparison with Answer Choices

Let's compare our final answer with the answer choices provided:

A. $2 q^{20}$

B. $2 q^{625}$

C. $16 q^{20}$

D. $16 q^9$

Our final answer, $16 q^{20}$, matches answer choice C.

Key Takeaways

• The power rule of exponents states that for any non-zero number $a$ and integers $m$ and $n$, the following rule holds: $\left(a^m\right)^n = a^{m \cdot n}$.
• When applying the power rule, we multiply the exponents when the base is the same.
• The expression $\left(2 q^5\right)^4$ can be simplified using the power rule to arrive at the final answer: $16 q^{20}$.

Additional Examples and Practice

To further practice and reinforce your understanding of exponents and power rules, try simplifying the following expressions:

• $\left(3 x^2\right)^3$
• $\left(4 y^4\right)^2$
• $\left(2 z^3\right)^5$

Use the power rule to simplify each expression and arrive at the final answer.
Simplify the Expression: $\left(2 q^5\right)^4$ - Q&A

Frequently Asked Questions

In our previous article, we simplified the expression $\left(2 q^5\right)^4$ using the power rule of exponents. However, we understand that some readers may still have questions or concerns about the topic. In this article, we will address some of the most frequently asked questions related to simplifying expressions with exponents.

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any non-zero number $a$ and integers $m$ and $n$, the following rule holds: $\left(a^m\right)^n = a^{m \cdot n}$. This rule allows us to simplify expressions by multiplying the exponents when the base is the same.

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

A: To apply the power rule, simply multiply the exponents when the base is the same. For example, if we have the expression $\left(2 q^5\right)^4$, we can simplify it by multiplying the exponents: $2^4 \cdot \left(q^5\right)^4 = 16 \cdot q^{20}$.

Q: What if the base is not the same?

A: If the base is not the same, we cannot apply the power rule. In this case, we need to use other rules or properties of exponents to simplify the expression.

Q: Can I simplify expressions with negative exponents?

A: Yes, we can simplify expressions with negative exponents using the power rule. For example, if we have the expression $\left(2 q^{-5}\right)^4$, we can simplify it by multiplying the exponents: $2^4 \cdot \left(q^{-5}\right)^4 = 16 \cdot q^{-20}$.

Q: How do I evaluate the exponent?

A: To evaluate the exponent, simply multiply the numbers together. For example, if we have the expression $q^{5 \cdot 4}$, we can evaluate the exponent by multiplying 5 and 4: $5 \cdot 4 = 20$.

Q: What if I have a fraction with exponents?

A: If you have a fraction with exponents, you can simplify it by applying the power rule to each part of the fraction separately. For example, if we have the expression $\frac{\left(2 q^5\right)^4}{\left(3 q^2\right)^3}$, we can simplify it by applying the power rule to each part: $\frac{16 \cdot q^{20}}{27 \cdot q^6}$.

Q: Can I simplify expressions with variables in the exponent?

A: Yes, we can simplify expressions with variables in the exponent using the power rule. For example, if we have the expression $\left(2 x^2\right)^4$, we can simplify it by multiplying the exponents: $2^4 \cdot \left(x^2\right)^4 = 16 \cdot x^8$.

Conclusion

In this article, we addressed some of the most frequently asked questions related to simplifying expressions with exponents. We hope that this article has provided you with a better understanding of the power rule and how to apply it to simplify expressions. If you have any further questions or concerns, please don't hesitate to ask.

Additional Resources

For more information on simplifying expressions with exponents, we recommend the following resources:

• Khan Academy: Exponents and Power Rules
• Mathway: Simplifying Expressions with Exponents
• Wolfram Alpha: Exponents and Power Rules

Practice Problems

To further practice and reinforce your understanding of exponents and power rules, try simplifying the following expressions:

• $\left(3 x^2\right)^3$
• $\left(4 y^4\right)^2$
• $\left(2 z^3\right)^5$

Use the power rule to simplify each expression and arrive at the final answer.