Simplify The Expression:${ \left(3 Y {-2}\right) 2 }$

Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. In this case, we're given the expression $\left(3 y^{-2}\right)^2$ and we need to simplify it. To start, let's recall the rule for raising a power to another power: $(a^m)^n = a^{m \cdot n}$. This rule will be crucial in simplifying the given expression.

Applying the Power Rule

Using the power rule, we can rewrite the expression as $3^2 \cdot (y^{-2})^2$. Now, let's simplify the exponent of $y$. When we raise a negative exponent to another power, we multiply the exponents. Therefore, $(y^{-2})^2 = y^{-2 \cdot 2} = y^{-4}$.

Simplifying the Expression

Now that we've simplified the exponent of $y$, we can rewrite the expression as $3^2 \cdot y^{-4}$. The exponent of $3$ is simply $2$, so we can rewrite the expression as $9 \cdot y^{-4}$.

Understanding Negative Exponents

Before we can simplify the expression further, let's take a closer look at negative exponents. A negative exponent indicates that we need to take the reciprocal of the base. In this case, we have $y^{-4}$, which means we need to take the reciprocal of $y$ and raise it to the power of $4$. This can be rewritten as $\frac{1}{y^4}$.

Simplifying the Expression Further

Now that we've understood negative exponents, we can simplify the expression further. We have $9 \cdot y^{-4}$, which is equivalent to $9 \cdot \frac{1}{y^4}$. To simplify this expression, we can multiply the numerator and denominator by $y^4$ to get rid of the fraction. This gives us $\frac{9 \cdot y^4}{y^4 \cdot y^4}$.

Canceling Out Common Factors

Now that we've multiplied the numerator and denominator by $y^4$, we can simplify the expression further by canceling out common factors. The $y^4$ terms in the numerator and denominator cancel out, leaving us with $\frac{9}{y^8}$.

Final Simplification

The expression $\frac{9}{y^8}$ is the final simplified form of the original expression $\left(3 y^{-2}\right)^2$. This expression indicates that the value of $y$ is being raised to the power of $8$ and then taking the reciprocal of it, and then multiplying it by $9$.

Conclusion

In this article, we've simplified the expression $\left(3 y^{-2}\right)^2$ using the power rule and understanding negative exponents. We've shown that the expression can be simplified to $\frac{9}{y^8}$, which indicates that the value of $y$ is being raised to the power of $8$ and then taking the reciprocal of it, and then multiplying it by $9$. This expression is a fundamental concept in mathematics and is used extensively in various fields, including physics and engineering.

Frequently Asked Questions

• What is the power rule in mathematics? The power rule states that when we raise a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• How do we simplify negative exponents? A negative exponent indicates that we need to take the reciprocal of the base. For example, $y^{-4}$ is equivalent to $\frac{1}{y^4}$.
• What is the final simplified form of the expression $\left(3 y^{-2}\right)^2$? The final simplified form of the expression is $\frac{9}{y^8}$.

Key Takeaways

• The power rule is a fundamental concept in mathematics that states that when we raise a power to another power, we multiply the exponents.
• Negative exponents indicate that we need to take the reciprocal of the base.
• The final simplified form of the expression $\left(3 y^{-2}\right)^2$ is $\frac{9}{y^8}$.

Further Reading

• For more information on the power rule, see Power Rule.
• For more information on negative exponents, see Negative Exponent.
• For more information on simplifying expressions, see Simplifying Expressions.
  

Frequently Asked Questions

We've received many questions about simplifying the expression $\left(3 y^{-2}\right)^2$. Here are some of the most frequently asked questions and their answers.

Q: What is the power rule in mathematics?

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

Q: How do we simplify negative exponents?

A: A negative exponent indicates that we need to take the reciprocal of the base. For example, $y^{-4}$ is equivalent to $\frac{1}{y^4}$.

Q: What is the final simplified form of the expression $\left(3 y^{-2}\right)^2$?

A: The final simplified form of the expression is $\frac{9}{y^8}$.

Q: Can you explain the concept of negative exponents in more detail?

A: A negative exponent indicates that we need to take the reciprocal of the base. For example, $y^{-4}$ is equivalent to $\frac{1}{y^4}$. This means that we need to flip the fraction and change the sign of the exponent.

Q: How do we handle exponents with the same base?

A: When we have exponents with the same base, we can add or subtract the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: Can you provide an example of how to simplify an expression using the power rule?

A: Let's consider the expression $(2x^3)^4$. Using the power rule, we can rewrite this expression as $2^4 \cdot (x^3)^4 = 16 \cdot x^{12}$.

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

A: A positive exponent indicates that we need to multiply the base by itself as many times as the exponent. For example, $x^3$ means $x \cdot x \cdot x$. A negative exponent indicates that we need to take the reciprocal of the base and multiply it by itself as many times as the exponent. For example, $x^{-3}$ means $\frac{1}{x} \cdot \frac{1}{x} \cdot \frac{1}{x}$.

Q: Can you explain the concept of exponents in more detail?

A: Exponents are a shorthand way of writing repeated multiplication. For example, $x^3$ means $x \cdot x \cdot x$. Exponents can be positive or negative, and they can be used to simplify complex expressions.

Additional Resources

If you're still having trouble understanding the concept of exponents or simplifying expressions, here are some additional resources that may help:

• Power Rule
• Negative Exponent
• Simplifying Expressions
• Exponents

Conclusion

We hope this Q&A article has helped you understand the concept of simplifying expressions using the power rule and negative exponents. If you have any further questions or need additional help, please don't hesitate to ask.

Key Takeaways

• The power rule states that when we raise a power to another power, we multiply the exponents.
• Negative exponents indicate that we need to take the reciprocal of the base.
• The final simplified form of the expression $\left(3 y^{-2}\right)^2$ is $\frac{9}{y^8}$.

Final Tips

• Make sure to understand the concept of exponents and how to simplify expressions using the power rule and negative exponents.
• Practice simplifying expressions using the power rule and negative exponents to build your confidence and skills.
• Don't be afraid to ask for help if you're struggling with a particular concept or expression.