Problem 2 - College Algebra Essentials

Question

Approximate each number using a calculator. Round your answer to three decimal places. $$3^{2.4}$$

Step-by-Step Solution

Answer

The approximate value of \(3^{2.4}\) rounded to three decimal places is \(17.491\).

Step 1: Compute the value

Use the calculator to evaluate \(3^{2.4}\).

Step 2: Round to three decimal places

This means that we only want three numbers or digits after the decimal point. So, if there are more, we will omit the rest of the digits, and if the digit after the third decimal place is greater than or equal to 5, we will add 1 to the third digit; otherwise, we will keep the number as it is.

Key Concepts

Approximating Numbers

When working with mathematical problems, particularly those involving complex calculations, it's not always practical to use exact numbers. That's where approximating numbers comes into play. Approximation involves finding a value that is close enough to the exact answer for our purposes, yet is simpler and easier to work with.

For example, if you're dealing with measurements in a real-world scenario, an approximation may be necessary due to measurement limitations. While performing the exercise of approximating the value of \(3^{2.4}\), the goal is to simplify the result to a manageable figure, in this case, to three decimal places. This process reduces complexity without significantly affecting accuracy, making the number more comprehensible and easier to use in further calculations.

Exponentiation

The operation of raising a number to a power is known as exponentiation. In our exercise \(3^{2.4}\), 3 is the base and 2.4 is the exponent. This represents a number, 3, multiplied by itself 2.4 times, which is not as straightforward as whole number exponents. Exponentiation can show rapid growth or decay in fields such as finance, physics, and engineering, thus understanding how to compute and approximate exponents is critical.

Although it's challenging to visualize what it means to multiply a number by itself a non-whole number of times, exponentiation with non-integer exponents corresponds to roots and logarithmic functions, concepts well-established in mathematics.

Calculator Usage

Calculators are ubiquitous tools in math for carrying out operations that are cumbersome or impractical to do by hand. For performing exponentiation of the form \(3^{2.4}\), a calculator quickly becomes invaluable.

When using a calculator for our exercise, it's important to know the correct sequence of buttons to press, which typically involves entering the base number (3), followed by the exponentiation key, often labeled as ‘^’ or ‘exp’, and then entering the exponent (2.4). After pressing the equals sign, the calculator displays the computed value which we can approximate or round as needed.

Significant Figures

In scientific and technical calculations, the concept of significant figures is fundamental. It helps to communicate the precision of a measurement or computation. Significant figures are the digits that carry meaning contributing to its precision. This excludes all leading zeros, middle zeros only when they are between other significant digits, and trailing zeros when they are after the decimal point and before a non-zero digit.

In the exercise, we round \(3^{2.4}\) to three decimal places, ensuring that the answer retains three significant digits after the decimal place. This rounding provides a balance between accuracy and readability, making sure the key figures that contribute to the number’s accuracy are retained.