Evaluating the Outer Exponent

Oct 13, 2025 by miodeya.com 80 views

Understanding Exponents

Exponents are a fundamental concept in mathematics that can be used to simplify complex expressions and solve equations. In this article, we will focus on evaluating the expression $\left(\left(-\frac{3}{2}\right)^2\right)^3$ . To do this, we need to understand the rules of exponents and how to apply them to simplify expressions.

The Order of Operations

Before we dive into evaluating the expression, it's essential to understand the order of operations. The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is as follows:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Evaluating the Expression

Now that we understand the order of operations, let's evaluate the expression $\left(\left(-\frac{3}{2}\right)^2\right)^3$ . To do this, we need to follow the order of operations.

First, we need to evaluate the expression inside the innermost parentheses, which is $\left(-\frac{3}{2}\right)^2$ . To do this, we need to square the fraction.

Squaring a Fraction

When we square a fraction, we need to square both the numerator and the denominator. In this case, the numerator is -3 and the denominator is 2.

$\left(-\frac{3}{2}\right)^2 = \frac{(-3)^2}{(2)^2} = \frac{9}{4}$

Now that we have evaluated the expression inside the innermost parentheses, we can move on to the next step.

Evaluating the Outer Exponent

The next step is to evaluate the outer exponent, which is $3$ . To do this, we need to raise the result of the inner expression to the power of 3.

$\left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3} = \frac{729}{64}$

Conclusion

In conclusion, the correct answer is $\frac{729}{64}$ . This is the result of evaluating the expression $\left(\left(-\frac{3}{2}\right)^2\right)^3$ using the rules of exponents and the order of operations.

Answer Key

A. $-\frac{729}{64}$ (Incorrect) B. $\frac{729}{64}$ (Correct) C. $-\frac{243}{32}$ (Incorrect) D. $\frac{243}{32}$ (Incorrect)

Final Thoughts

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about evaluating exponents. Whether you are a student or a teacher, these questions and answers will help you to better understand the rules of exponents and how to apply them to simplify complex expressions.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is as follows:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate an exponential expression?

A: To evaluate an exponential expression, you need to raise the base to the power of the exponent. For example, to evaluate the expression $2^3$ , you need to raise 2 to the power of 3, which is equal to 8.

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the reciprocal of the base is raised to a power. For example, $2^3$ is equal to 8, while $2^{-3}$ is equal to $\frac{1}{8}$ .

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

A: To evaluate an expression with multiple exponents, you need to follow the order of operations. First, evaluate any expressions inside parentheses, then evaluate any exponential expressions, and finally evaluate any multiplication and division operations from left to right.

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

A: When multiplying exponents with the same base, you need to add the exponents. For example, $2^3 \cdot 2^4$ is equal to $2^{3+4}$ , which is equal to $2^7$ .

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

A: When dividing exponents with the same base, you need to subtract the exponents. For example, $\frac{2^4}{2^3}$ is equal to $2^{4-3}$ , which is equal to $2^1$ .

Q: How do I evaluate an expression with a negative base and a positive exponent?

A: To evaluate an expression with a negative base and a positive exponent, you need to raise the absolute value of the base to the power of the exponent. For example, $(-2)^3$ is equal to $(-2)^3$ , which is equal to $-8$ .

Q: How do I evaluate an expression with a negative base and a negative exponent?

A: To evaluate an expression with a negative base and a negative exponent, you need to raise the absolute value of the base to the power of the absolute value of the exponent. For example, $(-2)^{-3}$ is equal to $(-2)^{-3}$ , which is equal to $\frac{1}{-8}$ , or $-\frac{1}{8}$ .

Conclusion

In conclusion, evaluating exponents can be a challenging task, but by following the rules of exponents and the order of operations, we can simplify complex expressions and solve equations. We hope that this article has provided you with a better understanding of how to evaluate exponents and has helped you to improve your math skills.

Answer Key

Q: What is the order of operations? A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression.

Q: How do I evaluate an exponential expression? A: To evaluate an exponential expression, you need to raise the base to the power of the exponent.

Q: What is the difference between a positive and negative exponent? A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the reciprocal of the base is raised to a power.

Q: How do I evaluate an expression with multiple exponents? A: To evaluate an expression with multiple exponents, you need to follow the order of operations.

Q: What is the rule for multiplying exponents with the same base? A: When multiplying exponents with the same base, you need to add the exponents.

Q: What is the rule for dividing exponents with the same base? A: When dividing exponents with the same base, you need to subtract the exponents.

Q: How do I evaluate an expression with a negative base and a positive exponent? A: To evaluate an expression with a negative base and a positive exponent, you need to raise the absolute value of the base to the power of the exponent.

Q: How do I evaluate an expression with a negative base and a negative exponent? A: To evaluate an expression with a negative base and a negative exponent, you need to raise the absolute value of the base to the power of the absolute value of the exponent.