Question

Use a calculator to evaluate each expression. Round your answer to three decimal places. \((-8.11)^{-4}\)

Short answer

0.000

Step-by-step solution

Input the Base into the Calculator

Enter the number \(-8.11\) into the calculator.

Apply the Exponent

Use the exponentiation function on the calculator to raise \(-8.11\) to the power of \(-4\).

Interpret the Result

The calculator will return a decimal number. Round this number to three decimal places.

Key concepts

Negative Exponents

Negative exponents can be tricky, but they are fascinating! When you see an expression like \((-8.11)^{-4}\), the negative exponent means you need to take the reciprocal of the base raised to the positive exponent. In other words, \((-8.11)^{-4}\) is the same as \(\frac{1}{(-8.11)^{4}}\). This means you're flipping the base, which turns a very small number into a very large one—or vice versa.
Here’s a quick breakdown of how negative exponents work:

• An expression like \(a^{-n}\) is equal to \(\frac{1}{a^{n}}\).
• It's helpful to remember that the negative sign in the exponent indicates a reciprocal.
• This is essential when solving many algebraic equations or functions.

Practicing these concepts can help you become very comfortable with negative exponents.

Rounding Decimals

Rounding decimals ensures your answer is easy to read and useful in real-world scenarios. When rounding to three decimal places, you're focusing on maintaining three digits after the decimal point.
Here’s a step-by-step guide to rounding decimals:

• Identify the number you need to round to—remember, three decimal places.
• Look at the fourth digit. This is the digit that will help you decide to round up or down.
• If the fourth digit is 5 or higher, you round the third digit up by one. If it’s 4 or lower, the third digit stays the same.

For example, if your calculated result is 0.0001385 and you need to round to three decimal places, the rounded number will be 0.000139.

Step-by-Step Problem Solving

Breaking down problems into clear, manageable steps is the best way to ensure you understand each part of the process. Let's consider the problem \((-8.11)^{-4}\), which we solved with a calculator.
Here’s a strategy for approaching such exercises:

• **Step 1: Understand the Problem**: Realize that the problem involves negative exponents.
  
• **Step 2: Use the Calculator**: Input \(-8.11\) into the calculator first.
• **Step 3: Apply the Exponent**: Raise the base to the power of \-4\ using the exponentiation function on the calculator.
• **Step 4: Interpret the Result**: Once the calculator gives you a decimal, round it to three decimal places as practiced earlier.

This methodical approach also helps in ensuring no steps are skipped and each part of the problem is addressed in order. Practice this method with different problems to become proficient at breaking complex problems into simpler steps.