Evaluate The Expression:$\[ -10 + (-7 + 1)^2 \\]

Introduction

In mathematics, expressions with exponents are a fundamental concept that can be used to simplify complex calculations. An exponent is a small number that is raised to the power of a variable or a constant. In this article, we will evaluate the expression $-10 + (-7 + 1)^2$ and provide a step-by-step guide on how to simplify it.

Understanding the Expression

The given expression is $-10 + (-7 + 1)^2$. To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate the expression inside the parentheses.
2. Exponents: Evaluate any exponents in the expression.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is $-7 + 1$. To evaluate this expression, we need to follow the order of operations:

1. Addition: $-7 + 1 = -6$

So, the expression inside the parentheses is equal to $-6$.

Step 2: Evaluate the Exponent

The exponent in the expression is $2$. To evaluate this exponent, we need to raise the value inside the parentheses to the power of $2$:

$(-6)^2 = 36$

So, the value of the exponent is $36$.

Step 3: Evaluate the Expression

Now that we have evaluated the expression inside the parentheses and the exponent, we can substitute these values back into the original expression:

$-10 + (-7 + 1)^2 = -10 + 36$

Step 4: Evaluate the Final Expression

Finally, we can evaluate the final expression by adding $-10$ and $36$:

$-10 + 36 = 26$

Conclusion

In this article, we evaluated the expression $-10 + (-7 + 1)^2$ using the order of operations (PEMDAS). We followed the steps outlined above to simplify the expression and arrived at the final answer of $26$. This example demonstrates the importance of following the order of operations when evaluating expressions with exponents.

Common Mistakes to Avoid

When evaluating expressions with exponents, there are several common mistakes to avoid:

• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when evaluating expressions with exponents.
• Not evaluating expressions inside parentheses first: Evaluate expressions inside parentheses before evaluating exponents.
• Not raising the value inside the parentheses to the power of the exponent: Make sure to raise the value inside the parentheses to the power of the exponent.

Real-World Applications

Expressions with exponents have numerous real-world applications in fields such as:

• Science: Exponents are used to describe the growth or decay of populations, chemical reactions, and physical phenomena.
• Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other complex systems.
• Finance: Exponents are used to calculate interest rates, investment returns, and other financial metrics.

Final Thoughts

Introduction

In our previous article, we evaluated the expression $-10 + (-7 + 1)^2$ using the order of operations (PEMDAS). In this article, we will provide a Q&A guide to help you better understand how to evaluate expressions with exponents.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponents in the expression next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, you need to follow the order of operations (PEMDAS). For example, if you have the expression $-7 + 1$, you would evaluate it as follows:

1. Addition: $-7 + 1 = -6$

So, the expression inside the parentheses is equal to $-6$.

Q: How do I evaluate exponents?

A: To evaluate exponents, you need to raise the value inside the parentheses to the power of the exponent. For example, if you have the expression $(-6)^2$, you would evaluate it as follows:

$(-6)^2 = 36$

So, the value of the exponent is $36$.

Q: What is the difference between a coefficient and an exponent?

A: A coefficient is a number that is multiplied by a variable or a constant, while an exponent is a small number that is raised to the power of a variable or a constant. For example, in the expression $3x^2$, the $3$ is a coefficient and the $2$ is an exponent.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you need to follow the order of operations (PEMDAS). For example, if you have the expression $-10 + (-7 + 1)^2$, you would evaluate it as follows:

1. Evaluate the expression inside the parentheses: $-7 + 1 = -6$
2. Evaluate the exponent: $(-6)^2 = 36$
3. Evaluate the final expression: $-10 + 36 = 26$

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:

• Not following the order of operations (PEMDAS)
• Not evaluating expressions inside parentheses first
• Not raising the value inside the parentheses to the power of the exponent
• Not simplifying the expression correctly

Q: How do I apply expressions with exponents in real-world situations?

A: Expressions with exponents have numerous real-world applications in fields such as:

• Science: Exponents are used to describe the growth or decay of populations, chemical reactions, and physical phenomena.
• Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other complex systems.
• Finance: Exponents are used to calculate interest rates, investment returns, and other financial metrics.

Conclusion

In conclusion, evaluating expressions with exponents requires a clear understanding of the order of operations (PEMDAS) and the ability to simplify complex calculations. By following the steps outlined above and avoiding common mistakes, you can simplify expressions with exponents and arrive at accurate solutions.