Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to approximate the number. (Round to three decimal places.)\(0.26^{-0.8}\)

Step-by-Step Solution

Answer

After following all these steps, the approximate value of \(0.26^{-0.8}\) when rounded to three decimal places should be obtained.

Step 1: Understanding Negative Exponents

Knowing how negative exponents work is essential. The negative exponent n in a^n refers to the reciprocal of the base raised to the exponent. So, \(a^{-n} = 1/a^n\). In this case, \(0.26^{-0.8} = 1/(0.26^{0.8})\).

Step 2: Computing the Expression

Use a calculator to calculate the expression. Make sure to follow the order of operations - Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS). First, calculate 0.26 raised to 0.8 power, then take its reciprocal.

Step 3: Approximation

Round the resulting number to the nearest three decimal places. Make sure to understand the rules of rounding - if the digit to the right of the third decimal place is 5 or more, round up the last digit included. If it's less than 5, leave the last digit included as it is.

Key Concepts

Calculator Usage

In today’s digital age, calculators are invaluable tools for solving complex math problems. When dealing with calculations involving negative exponents, like \(0.26^{-0.8}\), a calculator simplifies the task:

• First, enter the base number, in this case, 0.26.
• Next, use the exponent feature, often denoted by a button labeled '^' or a similar exponent key.
• Input the exponent value, which is -0.8 for this exercise.
• Finally, press the equal sign '=' to obtain the result.

Ensure you double-check your calculator settings to prevent errors. Adjust any mode settings from integer or fraction modes to decimal mode, which is necessary when working with non-integer values.
For precision, especially in learning contexts, it’s recommended to repeat the calculation to verify your result.

Exponentiation

Exponentiation involves raising a base number to the power of an exponent. It's essential to understand its rules, particularly when handling negative exponents.

A negative exponent such as in \(a^{-n}\), means taking the reciprocal of the base raised to the positive exponent: \(a^{-n} = \frac{1}{a^n}\). This turns the exponentiation process into a reciprocal calculation.
Let's break down the steps with \(0.26^{-0.8}\):

• First, calculate the positive exponentiation \(0.26^{0.8}\).
• Use the resulting value to find the reciprocal as per the negative exponent rule.

This foundational knowledge helps tackle more intricate exponents effortlessly. Whether you're using a calculator or performing manual calculations, understanding exponentiation ensures you handle numbers like pros.

Rounding Decimals

Rounding decimals accurately is crucial, especially when dealing with exercises that require this process, such as \(0.26^{-0.8}\) that need rounding to three decimal places.

Here’s how you round decimals precisely:

• Compute the exact value using a calculator.
• Look at the fourth decimal place. If this digit is 5 or greater, increase the third decimal place by one.
• If it’s less than 5, keep the third decimal place as is.

For example, if your calculator shows 3.4875, you round it to 3.488. This skill ensures your answers are both accurate and meet the problem’s requirements. Practicing rounding also enhances precision in estimating and verifying results in various math contexts.