Forgetting to evaluate the exponent first

Oct 13, 2025 by talynt.org 136 views

Understanding Exponents and Negative Numbers

In mathematics, exponents are a shorthand way of representing repeated multiplication. For example, $5^6$ means $5 \times 5 \times 5 \times 5 \times 5 \times 5$ . When we have a negative number raised to a power, we need to consider the order of operations and the properties of exponents.

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:

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.

Properties of Exponents

There are several properties of exponents that we need to know when working with negative numbers and exponents. These properties include:

Product of Powers: When we multiply two powers with the same base, we can add the exponents. For example, $a^m \times a^n = a^{m+n}$ .
Power of a Power: When we raise a power to another power, we can multiply the exponents. For example, $(a^m)^n = a^{m \times n}$ .
Negative Exponent: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$ .

Evaluating $-5^6$

Now that we have a good understanding of exponents and negative numbers, let's evaluate the expression $-5^6$ . Using the order of operations, we need to evaluate the exponent first.

$-5^6 = -(5 \times 5 \times 5 \times 5 \times 5 \times 5)$

Using the product of powers property, we can rewrite this as:

$-5^6 = -(5^6)$

Now, we can use the property of negative exponents to rewrite this as:

$-5^6 = -\frac{1}{5^6}$

Finally, we can evaluate the expression by calculating the value of $5^6$ and taking the reciprocal.

$5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15,625$

So, $-5^6 = -\frac{1}{15,625} = -\frac{1}{15625}$

Conclusion

In conclusion, the value of $-5^6$ is $-\frac{1}{15,625}$ . This is a negative number, and it is the result of evaluating the expression using the order of operations and the properties of exponents.

Answer

The correct answer is A. -15,625.

Why is this important?

Understanding exponents and negative numbers is important in mathematics because it allows us to evaluate complex expressions and solve problems involving variables and constants. It is also a fundamental concept in algebra and is used extensively in mathematics, science, and engineering.

Real-World Applications

Exponents and negative numbers have many real-world applications. For example, in finance, exponents are used to calculate compound interest and investment returns. In science, exponents are used to describe the growth and decay of populations and the behavior of physical systems. In engineering, exponents are used to design and optimize systems and to model complex phenomena.

Common Mistakes

When working with exponents and negative numbers, there are several common mistakes to avoid. These include:

Forgetting to evaluate the exponent first: When working with exponents, it is easy to forget to evaluate the exponent first. Make sure to follow the order of operations and evaluate the exponent before performing any other operations.
Not using the correct property of exponents: There are several properties of exponents that we need to know when working with negative numbers and exponents. Make sure to use the correct property of exponents to evaluate the expression.
Not checking the answer: When evaluating an expression, make sure to check the answer to ensure that it is correct. This will help you avoid making mistakes and ensure that you get the correct answer.

Conclusion

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:

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: What is the product of powers property?

A: The product of powers property states that when we multiply two powers with the same base, we can add the exponents. For example, $a^m \times a^n = a^{m+n}$ .

Q: What is the power of a power property?

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

Q: What is the negative exponent property?

A: The negative exponent property states that when we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$ .

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

A: To evaluate an expression with a negative exponent, you can use the negative exponent property to rewrite the expression as a positive exponent. For example, $a^{-m} = \frac{1}{a^m}$ .

Q: What is the difference between $-5^6$ and $-(5^6)$ ?

A: $-5^6$ means $-(5^6)$ , which is equal to $-15625$ . On the other hand, $-(5^6)$ means $-15625$ .

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by using the negative exponent property to rewrite the expression as a positive exponent.

Q: How do I know when to use the order of operations?

A: You should use the order of operations whenever you have multiple operations in an expression. This will help you to evaluate the expression correctly and avoid mistakes.

Q: What are some common mistakes to avoid when working with exponents and negative numbers?

A: Some common mistakes to avoid when working with exponents and negative numbers include:

Forgetting to evaluate the exponent first
Not using the correct property of exponents
Not checking the answer

Q: Why is it important to understand exponents and negative numbers?

A: Understanding exponents and negative numbers is important because it allows you to evaluate complex expressions and solve problems involving variables and constants. It is also a fundamental concept in algebra and is used extensively in mathematics, science, and engineering.

Q: Can you give me some examples of real-world applications of exponents and negative numbers?

A: Yes, here are some examples of real-world applications of exponents and negative numbers:

Finance: Exponents are used to calculate compound interest and investment returns.
Science: Exponents are used to describe the growth and decay of populations and the behavior of physical systems.
Engineering: Exponents are used to design and optimize systems and to model complex phenomena.

Conclusion

In conclusion, understanding exponents and negative numbers is an important concept in mathematics that has many real-world applications. By following the order of operations and using the correct properties of exponents, you can evaluate complex expressions and solve problems involving variables and constants. Remember to avoid common mistakes and to check your answers to ensure that you get the correct result.