Simplify The Expression:${ -8^2 \div (-2)^3 }$

Understanding the Expression

When simplifying the given expression, $-8^2 \div (-2)^3$, we need to follow the order of operations (PEMDAS/BODMAS) which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction. In this case, we have exponents and division, so we will start by evaluating the exponents.

Evaluating Exponents

The expression contains two exponents: $-8^2$ and $(-2)^3$. To evaluate these exponents, we need to follow the rules of exponentiation.

• $-8^2$ can be evaluated as $-(8^2)$, which is equal to $-64$.
• $(-2)^3$ can be evaluated as $-2 \times -2 \times -2$, which is equal to $-8$.

Simplifying the Expression

Now that we have evaluated the exponents, we can simplify the expression by dividing $-64$ by $-8$.

Division of Negative Numbers

When dividing two negative numbers, the result is always positive. Therefore, $-64 \div -8$ is equal to $8$.

Conclusion

In conclusion, the simplified expression $-8^2 \div (-2)^3$ is equal to $8$. This is because when we divide two negative numbers, the result is always positive.

Real-World Applications

Understanding how to simplify expressions involving exponents and division is crucial in various real-world applications, such as:

• Science and Engineering: In physics and engineering, exponents and division are used to calculate quantities such as force, energy, and velocity.
• Finance: In finance, exponents and division are used to calculate interest rates, investment returns, and stock prices.
• Computer Science: In computer science, exponents and division are used to calculate quantities such as memory usage, processing power, and data transfer rates.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions involving exponents and division:

• Follow the order of operations: Always follow the order of operations (PEMDAS/BODMAS) when simplifying expressions.
• Evaluate exponents first: Evaluate exponents before performing division.
• Remember the rules of exponentiation: Remember the rules of exponentiation, such as $a^m \times a^n = a^{m+n}$ and $(a^m)^n = a^{m \times n}$.
• Practice, practice, practice: Practice simplifying expressions involving exponents and division to become more comfortable with the concepts.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions involving exponents and division:

• Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.
• Not evaluating exponents first: Failing to evaluate exponents before performing division can lead to incorrect results.
• Not remembering the rules of exponentiation: Failing to remember the rules of exponentiation can lead to incorrect results.

Final Thoughts

In conclusion, simplifying the expression $-8^2 \div (-2)^3$ requires following the order of operations, evaluating exponents, and remembering the rules of exponentiation. By understanding these concepts and practicing regularly, you can become more comfortable with simplifying expressions involving exponents and division.

Understanding the Expression

When simplifying the given expression, $-8^2 \div (-2)^3$, we need to follow the order of operations (PEMDAS/BODMAS) which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction. In this case, we have exponents and division, so we will start by evaluating the exponents.

Q&A

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 simplifying an expression. The order of operations is:

1. Parentheses/Brackets
2. Exponents/Orders
3. Multiplication and Division
4. Addition and Subtraction

Q: How do I evaluate exponents?

A: To evaluate exponents, you need to follow the rules of exponentiation. The rules of exponentiation are:

• $a^m \times a^n = a^{m+n}$
• $(a^m)^n = a^{m \times n}$

Q: What is the difference between $-8^2$ and $-(8^2)$?

A: $-8^2$ is equal to $-(8^2)$, which is equal to $-64$. The negative sign is applied to the result of the exponentiation, not to the base.

Q: How do I simplify the expression $-8^2 \div (-2)^3$?

A: To simplify the expression $-8^2 \div (-2)^3$, you need to follow the order of operations. First, evaluate the exponents:

• $-8^2$ is equal to $-(8^2)$, which is equal to $-64$.
• $(-2)^3$ is equal to $-2 \times -2 \times -2$, which is equal to $-8$.

Then, divide $-64$ by $-8$.

Q: What is the result of $-64 \div -8$?

A: When dividing two negative numbers, the result is always positive. Therefore, $-64 \div -8$ is equal to $8$.

Q: What are some real-world applications of simplifying expressions involving exponents and division?

A: Simplifying expressions involving exponents and division is crucial in various real-world applications, such as:

• Science and Engineering: In physics and engineering, exponents and division are used to calculate quantities such as force, energy, and velocity.
• Finance: In finance, exponents and division are used to calculate interest rates, investment returns, and stock prices.
• Computer Science: In computer science, exponents and division are used to calculate quantities such as memory usage, processing power, and data transfer rates.

Q: What are some common mistakes to avoid when simplifying expressions involving exponents and division?

A: Some common mistakes to avoid when simplifying expressions involving exponents and division are:

• Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.
• Not evaluating exponents first: Failing to evaluate exponents before performing division can lead to incorrect results.
• Not remembering the rules of exponentiation: Failing to remember the rules of exponentiation can lead to incorrect results.

Final Thoughts

In conclusion, simplifying the expression $-8^2 \div (-2)^3$ requires following the order of operations, evaluating exponents, and remembering the rules of exponentiation. By understanding these concepts and practicing regularly, you can become more comfortable with simplifying expressions involving exponents and division.