Problem 9 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression to four decimal places using a calculator. $$4^{1.6}$$

Step-by-Step Solution

Answer

\(4^{1.6} \approx 9.1896\)

Step 1: Expression

Evaluate each expression to four decimal places using a calculator.
$$4^{1.6}$$

Step 2: Apply Rules

Apply appropriate mathematical rules step by step.

Step 3: Result

The value of \(4^{1.6}\) to four decimal places is approximately 6.9644

Key Concepts

Calculating Exponents

Understanding the concept of exponentiation is crucial for many areas of mathematics and science. Essentially, when we calculate an exponent, we are performing repeated multiplication. For example, the expression \(a^n\) means that you multiply the base \(a\) by itself \(n\) times, where \(n\) is the exponent. However, when the exponent is not a whole number, as in the expression \(4^{1.6}\), it refers to taking a root as well as a power.

Using a calculator simplifies the process, accurately handling complex calculations that involve decimals and irrational numbers. You can approximate \(4^{1.6}\) by considering it as \(4^{\frac{8}{5}}\), which involves finding the 5th root of 4 and then raising the result to the 8th power, but this can be cumbersome without a calculator. So, we input the expression exactly as it is presented into the calculator, pressing the exponentiation button after entering the base number 4, followed by 1.6 as the exponent.

Scientific Calculator Usage

A scientific calculator is a powerful tool for students and professionals working with complex mathematical concepts, including exponentiation. To compute something like \(4^{1.6}\), it's important to know how to navigate the calculator's functions. Most scientific calculators have a dedicated exponent button, often labeled as '^', 'exp', or displaying a 'y^x' symbol.

Select the base number (4 in this case), press the exponent button, then input the exponent value (1.6). Hit 'equals' or the equivalent button to get your result. It's important to familiarize yourself with the particular model of the calculator being used, as button placement and functions might vary. It may also be beneficial to refer to the calculator's manual for specific instructions on performing such operations. Modern calculators can display a wide range of digits, but in our case, we only concern ourselves with the first four decimal places.

Decimal Place Approximation

When we approximate to a certain number of decimal places, we are essentially rounding a number so that it has a specific number of digits after the decimal point. For our example, \(4^{1.6}\), we want to round the answer to four decimal places. This means we look to the fifth digit after the decimal point. If this digit is five or higher, we round up the fourth digit; if it's four or lower, we leave the fourth digit as it is.

Why is this important? In many practical applications, such as engineering or science, measurements and calculations can only be precise to a certain extent due to limitations in the tools used or variables involved in an experiment. Decimal place approximation allows us to present a number that is both accurate and practical. Always check the requirements given, as rounding correctly is vital in reporting and using your calculated results.