Simplify: ( − Y 4 ) 2 \left(-y^4\right)^2 ( − Y 4 ) 2

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Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. In this case, we're given the expression $\left(-y^4\right)^2$ and we need to simplify it. The main keywords in this problem are exponentiation, simplifying expressions, and mathematical operations.

Applying the Power Rule of Exponentiation

To simplify the given expression, we can use the power rule of exponentiation, which states that for any numbers $a$ and $b$ and any integer $n$, $\left(a^b\right)^n = a^{b \cdot n}$. This rule allows us to simplify expressions by multiplying the exponents.

Simplifying the Expression

Using the power rule, we can simplify the expression $\left(-y^4\right)^2$ as follows:

$\left(-y^4\right)^2 = (-1)^2 \cdot (y^4)^2$

Understanding the Properties of Exponents

When we raise a power to another power, we multiply the exponents. In this case, we have $(y^4)^2$, which means we need to multiply the exponent $4$ by $2$.

Simplifying the Expression Further

Using the property of exponents, we can simplify the expression further:

$(-1)^2 \cdot (y^4)^2 = (-1)^2 \cdot y^{4 \cdot 2}$

Evaluating the Expression

Now, we can evaluate the expression by simplifying the exponent:

$(-1)^2 \cdot y^{4 \cdot 2} = 1 \cdot y^8$

Final Answer

The final answer is $\boxed{y^8}$.

Understanding the Concept of Negative Exponents

In the previous example, we raised a negative number to an even power, which resulted in a positive number. However, if we raise a negative number to an odd power, the result will be negative.

Example: Simplifying $\left(-y^3\right)^2$

To simplify the expression $\left(-y^3\right)^2$, we can use the power rule of exponentiation:

$\left(-y^3\right)^2 = (-1)^2 \cdot (y^3)^2$

Simplifying the Expression

Using the property of exponents, we can simplify the expression further:

$(-1)^2 \cdot (y^3)^2 = (-1)^2 \cdot y^{3 \cdot 2}$

Evaluating the Expression

Now, we can evaluate the expression by simplifying the exponent:

$(-1)^2 \cdot y^{3 \cdot 2} = 1 \cdot y^6$

Final Answer

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

Understanding the Concept of Negative Exponents

In the previous examples, we raised negative numbers to even powers, which resulted in positive numbers. However, if we raise a negative number to an odd power, the result will be negative.

Example: Simplifying $\left(-y^4\right)^3$

To simplify the expression $\left(-y^4\right)^3$, we can use the power rule of exponentiation:

$\left(-y^4\right)^3 = (-1)^3 \cdot (y^4)^3$

Simplifying the Expression

Using the property of exponents, we can simplify the expression further:

$(-1)^3 \cdot (y^4)^3 = (-1)^3 \cdot y^{4 \cdot 3}$

Evaluating the Expression

Now, we can evaluate the expression by simplifying the exponent:

$(-1)^3 \cdot y^{4 \cdot 3} = -1 \cdot y^{12}$

Final Answer

The final answer is $\boxed{-y^{12}}$.

Conclusion

In this article, we simplified expressions involving negative numbers and exponents. We used the power rule of exponentiation to simplify expressions and evaluated the results. We also discussed the concept of negative exponents and how they affect the result of an expression.

Key Takeaways

• The power rule of exponentiation states that for any numbers $a$ and $b$ and any integer $n$, $\left(a^b\right)^n = a^{b \cdot n}$.
• When we raise a power to another power, we multiply the exponents.
• If we raise a negative number to an even power, the result will be positive.
• If we raise a negative number to an odd power, the result will be negative.

Final Thoughts

Simplifying expressions involving negative numbers and exponents requires a good understanding of the power rule of exponentiation and the properties of exponents. By following the steps outlined in this article, you can simplify complex expressions and evaluate the results.

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Frequently Asked Questions

Q: What is the power rule of exponentiation?

A: The power rule of exponentiation states that for any numbers $a$ and $b$ and any integer $n$, $\left(a^b\right)^n = a^{b \cdot n}$. This rule allows us to simplify expressions by multiplying the exponents.

Q: How do I simplify an expression with a negative number raised to an even power?

A: To simplify an expression with a negative number raised to an even power, you can use the power rule of exponentiation. For example, $\left(-y^4\right)^2 = (-1)^2 \cdot (y^4)^2 = 1 \cdot y^8$.

Q: How do I simplify an expression with a negative number raised to an odd power?

A: To simplify an expression with a negative number raised to an odd power, you can use the power rule of exponentiation. For example, $\left(-y^4\right)^3 = (-1)^3 \cdot (y^4)^3 = -1 \cdot y^{12}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as a reciprocal. For example, $x^3$ is a positive exponent, while $x^{-3}$ is a negative exponent.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the property of exponents that states $a^{-n} = \frac{1}{a^n}$. For example, $x^{-3} = \frac{1}{x^3}$.

Q: What is the rule for multiplying exponents with the same base?

A: The rule for multiplying exponents with the same base is that you add the exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$.

Q: What is the rule for dividing exponents with the same base?

A: The rule for dividing exponents with the same base is that you subtract the exponents. For example, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the rules for multiplying and dividing exponents. For example, $x^2 \cdot x^3 \cdot x^4 = x^{2+3+4} = x^9$.

Q: What is the rule for raising a power to another power?

A: The rule for raising a power to another power is that you multiply the exponents. For example, $(x^2)^3 = x^{2 \cdot 3} = x^6$.

Q: How do I simplify an expression with a negative number raised to a power and then raised to another power?

A: To simplify an expression with a negative number raised to a power and then raised to another power, you can use the power rule of exponentiation. For example, $\left(-y^4\right)^2 = (-1)^2 \cdot (y^4)^2 = 1 \cdot y^8$.

Conclusion

In this Q&A article, we answered some of the most frequently asked questions about simplifying expressions involving negative numbers and exponents. We covered topics such as the power rule of exponentiation, simplifying expressions with negative numbers raised to even and odd powers, and the rules for multiplying and dividing exponents.

Key Takeaways

• The power rule of exponentiation states that for any numbers $a$ and $b$ and any integer $n$, $\left(a^b\right)^n = a^{b \cdot n}$.
• When we raise a power to another power, we multiply the exponents.
• If we raise a negative number to an even power, the result will be positive.
• If we raise a negative number to an odd power, the result will be negative.
• The rule for multiplying exponents with the same base is that you add the exponents.
• The rule for dividing exponents with the same base is that you subtract the exponents.

Final Thoughts

Simplifying expressions involving negative numbers and exponents requires a good understanding of the power rule of exponentiation and the properties of exponents. By following the steps outlined in this article, you can simplify complex expressions and evaluate the results.