Applying the Rules of Exponents

Oct 13, 2025 by vinooalluri.com 17 views

Understanding the Problem

The given problem involves simplifying an algebraic expression that contains exponents and variables. We are required to simplify the expression $\left(-2 x^3\right)^2 \cdot y \cdot y^9$ .

Applying the Rules of Exponents

To simplify the expression, we need to apply the rules of exponents. The first step is to expand the squared term using the power rule of exponents, which states that $\left(a^m\right)^n = a^{m \cdot n}$ .

Expanding the Squared Term

Using the power rule of exponents, we can expand the squared term as follows:

$\left(-2 x^3\right)^2 = (-2)^2 \cdot (x^3)^2 = 4 \cdot x^{3 \cdot 2} = 4 \cdot x^6$

Simplifying the Expression

Now that we have expanded the squared term, we can simplify the expression by multiplying the terms together. We have:

$\left(-2 x^3\right)^2 \cdot y \cdot y^9 = 4 \cdot x^6 \cdot y \cdot y^9$

Applying the Product Rule of Exponents

To simplify the expression further, we need to apply the product rule of exponents, which states that $a^m \cdot a^n = a^{m + n}$ . We can apply this rule to the terms $y$ and $y^9$ as follows:

$y \cdot y^9 = y^{1 + 9} = y^{10}$

Simplifying the Expression

Now that we have applied the product rule of exponents, we can simplify the expression as follows:

$4 \cdot x^6 \cdot y \cdot y^9 = 4 \cdot x^6 \cdot y^{10}$

Final Answer

The final answer is $4 \cdot x^6 \cdot y^{10}$ , which can be written in the form $4 x^6 y^{10}$ .

Comparison with Answer Choices

We can compare our final answer with the answer choices provided:

A. $-4 x^6 y^8$ B. $-4 x^4 y^8$ C. $4 x^6 y^{10}$ D. $-4 x^4 y^{10}$

Our final answer, $4 x^6 y^{10}$ , matches answer choice C.

Conclusion

In this problem, we simplified the expression $\left(-2 x^3\right)^2 \cdot y \cdot y^9$ using the rules of exponents. We expanded the squared term, applied the product rule of exponents, and simplified the expression to obtain the final answer, $4 x^6 y^{10}$ .

Key Takeaways

The power rule of exponents states that $\left(a^m\right)^n = a^{m \cdot n}$ .
The product rule of exponents states that $a^m \cdot a^n = a^{m + n}$ .
To simplify an expression with exponents, we need to apply the rules of exponents and combine like terms.

Final Answer in Different Formats

The final answer is $4 x^6 y^{10}$ .
The final answer is $\boxed{4 x^6 y^{10}}$ .
The final answer is $4x^6y^{10}$ .
The final answer is $\boxed{4x^6y^{10}}$ .

Answer in Different Formats

The answer is C.
The answer is $\boxed{C}$ .
The answer is 4 x^6 y^{10}.
The answer is $\boxed{4 x^6 y^{10}}$ .

Frequently Asked Questions

Q: What is the power rule of exponents?

A: The power rule of exponents states that $\left(a^m\right)^n = a^{m \cdot n}$ . This means that when we raise a power to another power, we multiply the exponents.

Q: How do we apply the power rule of exponents?

A: To apply the power rule of exponents, we simply multiply the exponents together. For example, $\left(2^3\right)^2 = 2^{3 \cdot 2} = 2^6$ .

Q: What is the product rule of exponents?

A: The product rule of exponents states that $a^m \cdot a^n = a^{m + n}$ . This means that when we multiply two powers with the same base, we add the exponents.

Q: How do we apply the product rule of exponents?

A: To apply the product rule of exponents, we simply add the exponents together. For example, $2^3 \cdot 2^4 = 2^{3 + 4} = 2^7$ .

Q: How do we simplify an expression with exponents?

A: To simplify an expression with exponents, we need to apply the rules of exponents and combine like terms. We can use the power rule and product rule of exponents to simplify the expression.

Q: What is the final answer to the problem?

A: The final answer to the problem is $4 x^6 y^{10}$ .

Q: How do we compare the final answer with the answer choices?

A: To compare the final answer with the answer choices, we simply look at the expression and see which one matches. In this case, the final answer $4 x^6 y^{10}$ matches answer choice C.

Q: What are the key takeaways from this problem?

A: The key takeaways from this problem are:

The power rule of exponents states that $\left(a^m\right)^n = a^{m \cdot n}$ .
The product rule of exponents states that $a^m \cdot a^n = a^{m + n}$ .
To simplify an expression with exponents, we need to apply the rules of exponents and combine like terms.

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid when simplifying expressions with exponents include:

Not applying the power rule of exponents when raising a power to another power.
Not applying the product rule of exponents when multiplying two powers with the same base.
Not combining like terms when simplifying an expression.

Q: How do we check our work when simplifying expressions with exponents?

A: To check our work when simplifying expressions with exponents, we can plug in some values for the variables and see if the expression simplifies to the expected value. We can also use a calculator to check our work.

Additional Resources

For more information on the power rule of exponents, see Power Rule of Exponents.
For more information on the product rule of exponents, see Product Rule of Exponents.
For more information on simplifying expressions with exponents, see Simplifying Expressions with Exponents.

Conclusion

In this article, we have discussed how to simplify an expression with exponents using the power rule and product rule of exponents. We have also provided some common mistakes to avoid and some additional resources for further learning.