Using the Power of a Product Rule

Oct 13, 2025 by miodeya.com 101 views

Understanding the Problem

The problem requires us to find an equivalent expression to $\left(3 a b^2\right)^4$ for all positive values of $a$ and $b$ . This means we need to simplify the given expression and find a new expression that has the same value as the original expression for all positive values of $a$ and $b$ .

Using the Power of a Product Rule

To simplify the given expression, we can use the power of a product rule, which states that for any positive integers $m$ and $n$ , and any variables $x$ and $y$ :

$\left(xy\right)^m = x^my^m$

Using this rule, we can rewrite the given expression as:

$\left(3 a b^2\right)^4 = 3^4 a^4 b^{2 \cdot 4}$

Simplifying the Expression

Now, we can simplify the expression further by evaluating the powers:

$3^4 = 81$

$a^4 = a^4$

$b^{2 \cdot 4} = b^8$

So, the expression becomes:

$81 a^4 b^8$

Using the Power of a Power Rule

We can simplify the expression even further by using the power of a power rule, which states that for any positive integers $m$ and $n$ , and any variable $x$ :

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

Using this rule, we can rewrite the expression as:

$81 a^4 b^8 = \left(3^4\right) a^4 b^8 = 3^{4 \cdot 1} a^4 b^8 = 3^4 a^4 b^8$

Simplifying the Expression Again

Now, we can simplify the expression again by evaluating the powers:

$3^4 = 81$

$a^4 = a^4$

$b^8 = b^8$

So, the expression becomes:

$81 a^4 b^8$

Conclusion

In conclusion, the expression equivalent to $\left(3 a b^2\right)^4$ for all positive values of $a$ and $b$ is $81 a^4 b^8$ .

Final Answer

The final answer is $\boxed{81 a^4 b^8}$ .

Additional Information

The power of a product rule is a fundamental rule in algebra that allows us to simplify expressions involving products of variables.
The power of a power rule is another fundamental rule in algebra that allows us to simplify expressions involving powers of variables.
The rules of exponents are essential in algebra and are used to simplify expressions involving variables and their powers.

References

[1] Algebra, 2nd edition, by Michael Artin
[2] Calculus, 3rd edition, by Michael Spivak
[3] Algebra and Trigonometry, 4th edition, by James Stewart

Frequently Asked Questions

Q: What is the power of a product rule?

A: The power of a product rule is a fundamental rule in algebra that allows us to simplify expressions involving products of variables. It states that for any positive integers $m$ and $n$ , and any variables $x$ and $y$ :

$\left(xy\right)^m = x^my^m$

Q: How do I apply the power of a product rule to simplify the expression $\left(3 a b^2\right)^4$ ?

A: To simplify the expression, we can use the power of a product rule by rewriting the expression as:

$\left(3 a b^2\right)^4 = 3^4 a^4 b^{2 \cdot 4}$

Q: What is the power of a power rule?

A: The power of a power rule is another fundamental rule in algebra that allows us to simplify expressions involving powers of variables. It states that for any positive integers $m$ and $n$ , and any variable $x$ :

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

Q: How do I apply the power of a power rule to simplify the expression $81 a^4 b^8$ ?

A: To simplify the expression, we can use the power of a power rule by rewriting the expression as:

$81 a^4 b^8 = \left(3^4\right) a^4 b^8 = 3^{4 \cdot 1} a^4 b^8 = 3^4 a^4 b^8$

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{81 a^4 b^8}$ .

Q: What are some related problems that I can try?

A: Some related problems that you can try include:

Find an equivalent expression to $\left(2 x y^3\right)^5$ for all positive values of $x$ and $y$ .
Simplify the expression $\left(3 a b^2\right)^3$ for all positive values of $a$ and $b$ .
Find an equivalent expression to $\left(4 x^2 y^4\right)^2$ for all positive values of $x$ and $y$ .

Q: Where can I find more information on algebra and exponents?

A: You can find more information on algebra and exponents in the following resources:

Algebra, 2nd edition, by Michael Artin
Calculus, 3rd edition, by Michael Spivak
Algebra and Trigonometry, 4th edition, by James Stewart

Additional Tips and Tricks

Make sure to apply the power of a product rule and the power of a power rule correctly to simplify expressions involving variables and their powers.
Use the rules of exponents to simplify expressions involving variables and their powers.
Practice simplifying expressions involving variables and their powers to become more comfortable with the rules of exponents.

Conclusion

In conclusion, the expression equivalent to $\left(3 a b^2\right)^4$ for all positive values of $a$ and $b$ is $81 a^4 b^8$ . We hope that this Q&A article has been helpful in answering your questions and providing you with a better understanding of the rules of exponents.