Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the best decimal approximation that your calculator allows. Begin by making a mental estimate. \((3.1415)^{-1 / 2}\)

Step-by-Step Solution

Answer

Approximate value is 0.56419.

Step 1: Understand the Expression

The given expression is \((3.1415)^{-1 / 2}\). This expression is equivalent to the square root of the reciprocal of 3.1415.

Step 2: Interpret the Expression

Writing \((3.1415)^{-1 / 2}\) as a radical, we see it as \(\sqrt{1/3.1415}\). This means we first find the reciprocal of 3.1415 and then take the square root of that value.

Step 3: Calculate the Reciprocal

Find the reciprocal of 3.1415. Using a calculator, divide 1 by 3.1415, which gives approximately 0.31831.

Step 4: Compute the Square Root

Take the square root of the reciprocal obtained. Compute \(\sqrt{0.31831}\) using a calculator. This returns approximately 0.56419.

Step 5: Verify the Calculation

Double-check the calculations by repeating the steps on a calculator to ensure correctness: calculate the reciprocal again and find its square root.

Key Concepts

Reciprocal

A reciprocal is a simple yet crucial concept in mathematics. When we talk about the reciprocal of a number, we mean the value that, when multiplied by the original number, gives 1. To find the reciprocal of a number, you divide 1 by that number. For example, the reciprocal of 5 is \(\frac{1}{5}\), because 5 x \(\frac{1}{5}\) = 1.

Finding the reciprocal is essential in algebra and is often used in equations and expressions. In the context of our exercise, we began by calculating the reciprocal of 3.1415.

• The reciprocal of 3.1415 is calculated by dividing 1 by 3.1415, which gives approximately 0.31831.
• This step is necessary to transform our initial expression into a simpler form for further calculations, such as square roots or other operations.

Square Root

The square root of a number is another number that, when multiplied by itself, yields the original number. For example, the square root of 16 is 4 because 4 x 4 equals 16. Square roots provide a way to "undo" squaring and are denoted by the radical symbol \(\sqrt{}\).

In our exercise, after finding the reciprocal of 3.1415, we needed to find its square root.

• This involves calculating \(\sqrt{0.31831}\), which results in approximately 0.56419.
• Calculating square roots is a common operation when working with exponential expressions, as it helps simplify complex calculations.

Understanding square roots is critical for many areas of mathematics, including geometry and algebra.

Calculator Use

In modern mathematics, calculators are indispensable tools that help us solve complex problems quickly and accurately. Understanding how to effectively use a calculator can make mathematical computations much easier.

Let's take a closer look at how calculators can be used in solving expressions like our initial exercise:

• For calculating a reciprocal, enter the number and then divide 1 by this number.
• To find a square root, use the square root function, often labeled as \(\sqrt{}\) on your calculator.

When dealing with decimal approximations, always verify your results by re-entering the calculations. This ensures accuracy and helps confirm the correctness of your answer. Calculators not only provide a way to check your mental math but also offer insights into more complex mathematical concepts when used properly.