Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the expression. Round your result to three decimal places.\(100(1.03)^{-1.4}\)

Step-by-Step Solution

Answer

Upon performing the calculation, one gets the answer of approximately 78.422 when rounded off to three decimal places.

Step 1: Evaluate The Exponent

Begin by evaluating the power \(1.03^{-1.4}\). Use a scientific calculator, inputting 1.03 and pressing the power button (usually represented by '^'), then inputting '-1.4'.

Step 2: Carry Out The Multiplication

Next, multiply the result from step 1 by 100. This should give the final answer.

Step 3: Round-off To Three Decimal Places

Finally, round off the result from step 2 to three decimal places. This is done by looking at the fourth decimal place. If this figure is 5 or more, increase the third decimal place by one. If it is less than 5, leave the third decimal place as it is.

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number, called the base, to a power, also known as the exponent. In the expression given, which is \(1.03^{-1.4}\), 1.03 is our base and -1.4 is the exponent. The negative sign in the exponent indicates that we are dealing with a reciprocal. Thus, this operation essentially involves computing the reciprocal of the base raised to the absolute value of the exponent.

To calculate \(1.03^{-1.4}\), you need to perform the exponentiation in two steps: first raise 1.03 to the power of 1.4, and then take the reciprocal of this result. This kind of calculation often results in a non-integer value, which is why a calculator can be extremely helpful for precise results.

Scientific Calculator

A scientific calculator is an essential tool in evaluating complex mathematical functions like exponentiation. It includes specific keys for handling powers, logarithms, trigonometric functions, and other operations. When dealing with expressions like \(1.03^{-1.4}\), a scientific calculator simplifies the process greatly.

To use a scientific calculator for this task, start by entering the base (1.03), followed by the power or exponent key (often denoted by '^' or 'EXP'). Then, input the exponent value (-1.4). The calculator then computes the result efficiently and accurately.

• Make sure your calculator is in the correct mode before entering the numbers.
• Double-check the input values to ensure they're correct.
• Review the calculator's manual if you're unsure about any functions.

The calculator will give a precise result, which can then be adjusted as needed, such as for rounding.

Rounding Numbers

Rounding numbers is a common practice to simplify figures, making them easier to interpret and use, especially in scientific and academic contexts. In rounding to three decimal places, you look specifically at the fourth decimal place to decide whether to round the third figure up or leave it as it is.

Here's how to round a number to three decimal places:

• Identify the digit in the fourth decimal place.
• If this digit is 5 or greater, increase the number in the third decimal place by one.
• If it's less than 5, leave the third decimal place unchanged.

When you complete a calculation, like multiplying \(1.03^{-1.4}\) by 100, rounding the result to three decimal places may be necessary to fit the precision requirements of your exercise or problem. These steps ensure accuracy while maintaining simplicity in your final answer.