Problem 64 - A Graphical Approach to College Algebra

Question

Find a decimal approximation of each root or power. Round answers to the nearest thousandth. $$23^{2.75}$$

Step-by-Step Solution

Answer

The decimal approximation of \(23^{2.75}\) is 4865.295.

Step 1: Understanding the Expression

The expression given is \(23^{2.75}\). Our task is to calculate this power and find its decimal approximation.

Step 2: Calculating the Exponent

To calculate \(23^{2.75}\), it is equivalent to taking the base \(23\) and raising it to the power of \(2.75\). This can be divided into two parts: first finding \(23^2\), and then multiplying the result by \(23^{0.75}\).

Step 3: Calculating the Integer Power

Start by calculating \(23^2\), which is equal to \(529\).

Step 4: Calculating the Fractional Power

Now, calculate \(23^{0.75}\). By using a calculator, you will find that \(23^{0.75} \approx 9.193\).

Step 5: Multiplying the Results Together

Multiply the results of the integer power and the fractional power: \(529 \times 9.193 \approx 4865.295\).

Step 6: Rounding the Result

Round \(4865.295\) to the nearest thousandth, which gives you \(4865.295\).

Key Concepts

Exponents

An exponent tells you how many times to multiply a number by itself. In the expression \(23^{2.75}\), 23 is the base, and 2.75 is the exponent. This means we are multiplying the number 23 by itself a fraction of times. It's important to understand a few basics about exponents:

• Whole Number Exponents: When the exponent is a whole number, like 2 in \(23^2\), you simply multiply the base by itself. \(23^2 = 23 \times 23 = 529\).
• Zero Exponent: Any number raised to the power of zero is 1. For example, \(23^0 = 1\).
• Fractional Exponents: These exponents are not whole numbers and can represent roots of numbers as well, which will be discussed in the next section.

The key takeaway is that exponents simplify repeated multiplication, making calculations easier and more efficient. Understanding how to work with exponents also lays the foundation for tackling more complex mathematical problems.

Fractional Powers

Fractional powers might seem complex, but they are simply a different way to represent roots. The fractional exponent \(23^{0.75}\) involves both multiplication and root operations.

• Understanding the Fraction: The fraction 0.75 can be expressed as \(\frac{3}{4}\). This means that \(23^{0.75}\) is the same as \(\sqrt[4]{23^3}\).
• Calculating Fractional Exponents: When working with fractional exponents, it often involves using a calculator, as they can be more challenging to compute manually.
• Using Calculators: Modern calculators can handle these computations effortlessly. Simply enter the base followed by the power to obtain the result. For example, when you enter \(23^{0.75}\), you find approximately 9.193.

Fractional powers expand our mathematical toolbox, allowing us to express and compute solutions for a variety of problems beyond basic whole number powers.

Rounding Numbers

Rounding numbers helps to simplify complex calculations, particularly when dealing with many decimal places. In our solution, rounding makes the result more comprehensible.

• Rounding to the Nearest Thousandth: To round a number to the nearest thousandth, look at the digit in the ten-thousandth place (the fourth decimal). If it is 5 or greater, round up the third decimal place. If it is less than 5, leave the third decimal place as is.
• An Example: In our case, 4865.295 is rounded to 4865.295 because the digit in the ten-thousandth place is a 5. Thus, we don’t alter the third decimal place.
• Why Round? Rounding can make numbers easier to communicate and understand. It’s particularly useful when presenting results with many decimals that aren't necessary for the context.

Understanding rounding is crucial for ensuring precision and clarity in mathematical problems, especially ones involving long decimal strings.