Simplify ( − 2 P 2 Q 3 ) 4 \left(-2 P^2 Q^3\right)^4 ( − 2 P 2 Q 3 ) 4 .A. 16 P 8 Q 12 16 P^8 Q^{12} 16 P 8 Q 12 B. − 2 P 6 Q 7 -2 P^6 Q^7 − 2 P 6 Q 7 C. − 16 P 6 Q 7 -16 P^6 Q^7 − 16 P 6 Q 7 D. − 2 P 8 Q 12 -2 P^8 Q^{12} − 2 P 8 Q 12

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Understanding the Problem

When dealing with exponents, we need to remember the rules of exponentiation. The first rule is that when we raise a power to another power, we multiply the exponents. This is known as the power rule of exponents. The second rule is that when we have a negative exponent, we can rewrite it as a positive exponent by moving the base to the other side of the fraction bar. This is known as the negative exponent rule.

Applying the Power Rule of Exponents

To simplify the expression $\left(-2 p^2 q^3\right)^4$, we need to apply the power rule of exponents. This rule states that when we raise a power to another power, we multiply the exponents. In this case, we have $\left(-2 p^2 q^3\right)^4$, which means we need to multiply the exponent of $-2$, $p^2$, and $q^3$ by $4$.

Simplifying the Expression

Using the power rule of exponents, we can simplify the expression as follows:

$\left(-2 p^2 q^3\right)^4 = (-2)^4 \cdot (p^2)^4 \cdot (q^3)^4$

Evaluating the Exponents

Now, we need to evaluate the exponents. The exponent of $-2$ is $4$, which means we need to raise $-2$ to the power of $4$. The exponent of $p^2$ is also $4$, which means we need to raise $p^2$ to the power of $4$. The exponent of $q^3$ is also $4$, which means we need to raise $q^3$ to the power of $4$.

Simplifying the Terms

Using the rules of exponentiation, we can simplify the terms as follows:

$(-2)^4 = 16$

$(p^2)^4 = p^{2 \cdot 4} = p^8$

$(q^3)^4 = q^{3 \cdot 4} = q^{12}$

Combining the Terms

Now, we can combine the terms to simplify the expression:

$\left(-2 p^2 q^3\right)^4 = 16 \cdot p^8 \cdot q^{12}$

Final Answer

The final answer is $\boxed{16 p^8 q^{12}}$.

Comparison with the Options

Let's compare the final answer with the options:

A. $16 p^8 q^{12}$

B. $-2 p^6 q^7$

C. $-16 p^6 q^7$

D. $-2 p^8 q^{12}$

The final answer matches option A.

Conclusion

In this article, we simplified the expression $\left(-2 p^2 q^3\right)^4$ using the power rule of exponents. We evaluated the exponents and simplified the terms to arrive at the final answer. The final answer is $\boxed{16 p^8 q^{12}}$, which matches option A.

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Frequently Asked Questions

Q: What is the power rule of exponents?

A: The power rule of exponents states that when we raise a power to another power, we multiply the exponents. This means that if we have an expression like $\left(a^m\right)^n$, we can simplify it to $a^{m \cdot n}$.

Q: How do we apply the power rule of exponents to the expression $\left(-2 p^2 q^3\right)^4$?

A: To apply the power rule of exponents, we need to multiply the exponents of $-2$, $p^2$, and $q^3$ by $4$. This means that we need to evaluate $(-2)^4$, $(p^2)^4$, and $(q^3)^4$.

Q: What is the value of $(-2)^4$?

A: The value of $(-2)^4$ is $16$. This is because when we raise a negative number to an even power, the result is always positive.

Q: What is the value of $(p^2)^4$?

A: The value of $(p^2)^4$ is $p^8$. This is because when we raise a power to another power, we multiply the exponents.

Q: What is the value of $(q^3)^4$?

A: The value of $(q^3)^4$ is $q^{12}$. This is because when we raise a power to another power, we multiply the exponents.

Q: How do we combine the terms to simplify the expression?

A: To combine the terms, we multiply the values of $(-2)^4$, $(p^2)^4$, and $(q^3)^4$. This gives us $16 \cdot p^8 \cdot q^{12}$.

Q: What is the final answer?

A: The final answer is $\boxed{16 p^8 q^{12}}$.

Q: How does the final answer match with the options?

A: The final answer matches option A.

Q: What is the significance of the power rule of exponents in simplifying expressions?

A: The power rule of exponents is a fundamental rule in algebra that helps us simplify expressions by multiplying the exponents. It is a powerful tool that can be used to simplify complex expressions and make them easier to work with.

Q: Can the power rule of exponents be used to simplify expressions with negative exponents?

A: Yes, the power rule of exponents can be used to simplify expressions with negative exponents. When we raise a power to another power, we multiply the exponents, regardless of whether the exponent is positive or negative.

Q: How do we handle negative exponents when applying the power rule of exponents?

A: When we have a negative exponent, we can rewrite it as a positive exponent by moving the base to the other side of the fraction bar. This means that if we have an expression like $\left(\frac{a}{b}\right)^n$, we can rewrite it as $\frac{a^n}{b^n}$.

Q: What is the final answer to the expression $\left(-2 p^2 q^3\right)^4$?

A: The final answer to the expression $\left(-2 p^2 q^3\right)^4$ is $\boxed{16 p^8 q^{12}}$.

Conclusion

In this article, we answered some frequently asked questions about simplifying the expression $\left(-2 p^2 q^3\right)^4$ using the power rule of exponents. We covered topics such as the power rule of exponents, applying the power rule to the expression, evaluating the exponents, combining the terms, and handling negative exponents. We also provided the final answer and compared it with the options.