Problem 23 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the exponential function when \(x=2.5 .\) Round your answer to the nearest hundredth. $$y=5^{x}$$

Step-by-Step Solution

Answer

After rounding to the nearest hundredth, the answer is approximately 55.90.

Step 1: Input the Function into a Calculator

Using a scientific calculator, we will input the function. We'll type '5', then use the exponent key (usually marked as '^' or 'EXP'), and finally input '2.5'. The calculator will now show the result for \(5^{2.5}\).

Step 2: Rounding the Result

The calculator will display the precise answer, however, we're asked to round our result to the nearest hundredth. This means we need to look at the number in the thousandths place (the third decimal), and if that number is 5 or greater, we increase the value in the hundredths place (the second decimal) by 1. Conversely, if it is 4 or less, we leave the hundredths place unchanged. After that, we take the number up to the second decimal point only, ignoring the rest.

Key Concepts

Scientific Calculator

A scientific calculator is a handy tool for solving complex mathematical problems. It can perform standard operations like addition and multiplication, but it also excels at evaluating exponential functions. To use it, first locate the keys for base numbers and exponential notation, typically marked as `^` or `EXP`.

When you need to compute an expression like \(5^{2.5}\), you simply input the base number '5', press the exponent key, and then enter '2.5'. The calculator automatically processes this and provides an immediate result. This saves time and ensures accuracy compared to manual calculations.

Scientific calculators make tackling assignments involving exponential functions much more manageable, particularly when they require high precision.

Rounding Numbers

Rounding numbers helps simplify complex results to make them easier to understand and apply to real-world problems. When rounding, you convert a number to a nearby value with fewer digits. This guide will walk you through the process of rounding to the nearest hundredth.

If you're rounding a number like 76.843 to the nearest hundredth, first identify the digit in the thousandths place, which is '3'. The rules for rounding are:

• If the thousandths digit is 5 or greater, increase the hundredths digit by one.
• If it's 4 or less, leave the hundredths digit as is.

Therefore, 76.843 becomes 76.84 as it's already rounded down.

This step-by-step approach ensures your results conform to desired precision levels, which is essential in areas requiring exact figures, such as science and finance.

Evaluating Expressions

Evaluating expressions means breaking down mathematical problems to their simplest form to find a solution. With exponential functions like \(y=5^x\), this involves knowing your base (5) and your exponent (2.5) to compute the value of \(y\).

Let's illustrate the process for \(5^{2.5}\):

• First, enter the base '5' into your calculator.
• Use the exponent function key and input '2.5'.
• The calculator then evaluates \(5^{2.5}\) and provides the result.

The final number you get is the evaluated expression, which represents \(y\) for the given \(x\) value.

This method helps in solving parameters in various fields, especially when dealing with large datasets and requiring quick computations.