Exponents And Decimals Solution: -4² + (-3)³ + 11² + 126⁰

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Hey guys! Let's dive into solving this math problem step-by-step. We need to figure out the result of the expression: -4² + (-3)³ + 11² + 126⁰. This involves understanding exponents and how to deal with negative numbers and zero exponents. So, grab your pencils, and let’s get started!

Breaking Down the Expression

First, let's look at each term individually to make sure we understand what they mean:

  • -4²: Here, only the 4 is squared, and then we apply the negative sign. So, it's -(4 * 4) = -16.
  • (-3)³: This means -3 multiplied by itself three times: (-3) * (-3) * (-3) = -27.
  • 11²: This is 11 squared, which is 11 * 11 = 121.
  • 126⁰: Any non-zero number raised to the power of 0 is 1. So, 126⁰ = 1.

Now that we've broken down each part, the expression looks like this: -16 + (-27) + 121 + 1.

Step-by-Step Calculation

Let's piece it all together:

  1. Start with -16 + (-27). This is the same as -16 - 27. Adding two negative numbers gives us a more negative number. So, -16 - 27 = -43.
  2. Now we have: -43 + 121 + 1.
  3. Next, let’s add -43 and 121. This is the same as 121 - 43. When we subtract 43 from 121, we get 78. So, -43 + 121 = 78.
  4. Finally, we have 78 + 1. Adding these together gives us 79. So, 78 + 1 = 79.

Therefore, the result of the expression -4² + (-3)³ + 11² + 126⁰ is 79.

Detailed Explanation of Exponents

Alright, let’s deep-dive into exponents to make sure we're all on the same page. An exponent indicates how many times a number (the base) is multiplied by itself. For example, in the term a^b, a is the base, and b is the exponent.

  • Positive Integer Exponents: When the exponent is a positive integer, it's straightforward. For instance, 2^3 means 2 * 2 * 2, which equals 8.
  • Negative Exponents: A negative exponent means you take the reciprocal of the base raised to the positive exponent. For example, 2^(-3) is the same as 1 / (2^3) = 1 / 8 = 0.125.
  • Zero Exponent: Any non-zero number raised to the power of 0 is 1. This might seem weird, but it's a fundamental rule in mathematics. For example, 5^0 = 1, and (-3)^0 = 1.
  • Fractional Exponents: Fractional exponents involve roots. For instance, a^(1/2) is the square root of a, and a^(1/3) is the cube root of a. For example, 9^(1/2) = 3 because the square root of 9 is 3.

Understanding these rules is super important for solving expressions with exponents correctly!

Common Mistakes to Avoid

When dealing with exponents, it's easy to make a few common mistakes. Here are some pitfalls to watch out for:

  • Incorrectly Applying Negative Signs: Be careful with negative signs, especially when they're inside parentheses. Remember that -a² is different from (-a)². In the first case, you're squaring a and then applying the negative sign. In the second case, you're squaring -a.
  • Forgetting the Zero Exponent Rule: Many people forget that any non-zero number to the power of 0 is 1. This can lead to incorrect calculations.
  • Misunderstanding Negative Exponents: Negative exponents don't make the number negative; they represent the reciprocal of the base raised to the positive exponent.
  • Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Exponents come before multiplication, division, addition, and subtraction.

By being aware of these common mistakes, you can avoid errors and solve exponent problems more accurately.

Practice Problems

To solidify your understanding, let's tackle a few practice problems:

  1. Evaluate: -5² + (-2)⁴ + 7⁰
  2. Simplify: 3^(-2) + 4² - 10⁰
  3. Calculate: (-1)¹⁰ + (-1)¹¹ + (-1)¹²

Solutions:

  1. -5² + (-2)⁴ + 7⁰ = -25 + 16 + 1 = -8
  2. 3^(-2) + 4² - 10⁰ = 1/9 + 16 - 1 = 15 + 1/9 = 136/9
  3. (-1)¹⁰ + (-1)¹¹ + (-1)¹² = 1 + (-1) + 1 = 1

Work through these problems and check your answers. The more you practice, the more comfortable you'll become with exponents.

Real-World Applications

You might wonder, where do exponents show up in the real world? Well, they're everywhere!

  • Computer Science: Exponents are fundamental in computer science. For example, binary code (0s and 1s) uses powers of 2. The amount of memory in computers is often measured in powers of 2 (e.g., kilobytes, megabytes, gigabytes).
  • Finance: Compound interest involves exponents. When you earn interest on your savings, and that interest also earns interest, you're dealing with exponential growth.
  • Science: Exponential growth and decay are used to model population growth, radioactive decay, and many other phenomena. For example, the decay of radioactive isotopes is modeled using exponential functions.
  • Engineering: Exponents are used in various engineering calculations, such as determining the power of an engine or calculating the strength of a material.

Understanding exponents can help you make sense of the world around you and solve real-world problems.

Conclusion

So, to wrap things up, the answer to our original problem, -4² + (-3)³ + 11² + 126⁰, is 79. Remember, pay close attention to negative signs, understand the rules of exponents, and practice regularly. With these tips in mind, you'll be able to tackle any exponent problem with confidence. Keep up the great work, and you'll be an exponent expert in no time! Math can be fun, especially when you break it down step by step. Keep practicing, and you'll get better and better. You got this!