Evaluate The Expression { (3a)^{-2}$}$.
Introduction
In mathematics, expressions involving exponents and variables are common and can be challenging to evaluate. The expression {(3a)^{-2}$}$ is a good example of such an expression. In this article, we will evaluate this expression step by step, using the rules of exponents and algebra.
Understanding the Expression
The given expression is {(3a)^{-2}$}$. This expression involves a variable {a$}$ and a coefficient ${3\$}. The exponent indicates that the expression is being raised to the power of .
Rule of Negative Exponents
To evaluate this expression, we need to recall the rule of negative exponents. According to this rule, if a variable or expression is raised to a negative power, we can rewrite it as the reciprocal of the expression raised to the positive power. In other words, {a^{-n} = \frac{1}{a^n}$}$.
Applying the Rule of Negative Exponents
Using the rule of negative exponents, we can rewrite the given expression as:
{(3a)^{-2} = \frac{1}{(3a)^2}$}$
Evaluating the Expression Inside the Parentheses
Now, we need to evaluate the expression inside the parentheses, which is {(3a)^2$}$. To do this, we need to recall the rule of exponents, which states that when we raise a product to a power, we can raise each factor to that power. In other words, {(ab)^n = a^n \cdot b^n$}$.
Applying the Rule of Exponents
Using the rule of exponents, we can rewrite the expression inside the parentheses as:
{(3a)^2 = 3^2 \cdot a^2 = 9a^2$}$
Substituting the Value Back into the Original Expression
Now, we can substitute the value of the expression inside the parentheses back into the original expression:
{(3a)^{-2} = \frac{1}{(3a)^2} = \frac{1}{9a^2}$}$
Conclusion
In this article, we evaluated the expression {(3a)^{-2}$}$ step by step, using the rules of exponents and algebra. We first applied the rule of negative exponents to rewrite the expression as the reciprocal of the expression raised to the positive power. Then, we evaluated the expression inside the parentheses using the rule of exponents. Finally, we substituted the value back into the original expression to get the final result.
Final Answer
The final answer to the expression {(3a)^{-2}$}$ is {\frac{1}{9a^2}$}$.
Related Topics
- Exponents and Variables
- Rules of Exponents
- Algebraic Manipulations
Further Reading
- Khan Academy: Exponents and Variables
- Mathway: Exponents and Variables
- Wolfram Alpha: Exponents and Variables
References
- [1] "Algebra" by Michael Artin
- [2] "Calculus" by Michael Spivak
- [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
Introduction
In our previous article, we evaluated the expression {(3a)^{-2}$}$ step by step, using the rules of exponents and algebra. However, we know that there are many questions and doubts that readers may have regarding this topic. In this article, we will address some of the most frequently asked questions and provide additional explanations to help clarify any confusion.
Q&A
Q: What is the rule of negative exponents?
A: The rule of negative exponents states that if a variable or expression is raised to a negative power, we can rewrite it as the reciprocal of the expression raised to the positive power. In other words, {a^{-n} = \frac{1}{a^n}$}$.
Q: How do I apply the rule of negative exponents to the expression {(3a)^{-2}$}$?
A: To apply the rule of negative exponents, we need to rewrite the expression as the reciprocal of the expression raised to the positive power. In this case, we can rewrite the expression as:
{(3a)^{-2} = \frac{1}{(3a)^2}$}$
Q: What is the rule of exponents?
A: The rule of exponents states that when we raise a product to a power, we can raise each factor to that power. In other words, {(ab)^n = a^n \cdot b^n$}$.
Q: How do I apply the rule of exponents to the expression {(3a)^2$}$?
A: To apply the rule of exponents, we need to raise each factor to the power of 2. In this case, we can rewrite the expression as:
{(3a)^2 = 3^2 \cdot a^2 = 9a^2$}$
Q: What is the final answer to the expression {(3a)^{-2}$}$?
A: The final answer to the expression {(3a)^{-2}$}$ is {\frac{1}{9a^2}$}$.
Q: Can I use a calculator to evaluate the expression {(3a)^{-2}$}$?
A: Yes, you can use a calculator to evaluate the expression {(3a)^{-2}$}$. However, it's always a good idea to understand the underlying math and be able to evaluate the expression by hand.
Q: What are some common mistakes to avoid when evaluating expressions with exponents?
A: Some common mistakes to avoid when evaluating expressions with exponents include:
- Forgetting to apply the rule of negative exponents
- Forgetting to apply the rule of exponents
- Not simplifying the expression after applying the rules
- Not checking the final answer for errors
Conclusion
In this article, we addressed some of the most frequently asked questions and provided additional explanations to help clarify any confusion regarding the expression {(3a)^{-2}$}$. We hope that this article has been helpful in understanding the rules of exponents and algebra.
Final Answer
The final answer to the expression {(3a)^{-2}$}$ is {\frac{1}{9a^2}$}$.
Related Topics
- Exponents and Variables
- Rules of Exponents
- Algebraic Manipulations
Further Reading
- Khan Academy: Exponents and Variables
- Mathway: Exponents and Variables
- Wolfram Alpha: Exponents and Variables
References
- [1] "Algebra" by Michael Artin
- [2] "Calculus" by Michael Spivak
- [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton