Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the best decimal approximation that your calculator allows. Begin by making a mental estimate. \(\sqrt[4]{1.123}-\sqrt[3]{1.09}\)

Step-by-Step Solution

Answer

0.0004678

Step 1: Understand the Problem

We need to find the decimal approximation of the expression \( \sqrt[4]{1.123} - \sqrt[3]{1.09} \). This involves calculating the fourth root of 1.123 and the cube root of 1.09, then subtracting the latter from the former.

Step 2: Make a Mental Estimate

Estimate \( \sqrt[4]{1.123} \) and \( \sqrt[3]{1.09} \). Since both numbers are slightly greater than 1, their roots will be slightly greater than 1 too. For mental estimates, you can approximate \( \sqrt[4]{1.123} \approx 1.03 \) and \( \sqrt[3]{1.09} \approx 1.03 \). The difference should be very small.

Step 3: Calculate \( \sqrt[4]{1.123} \)

Use a calculator to find \( \sqrt[4]{1.123} \approx 1.0297948 \).

Step 4: Calculate \( \sqrt[3]{1.09} \)

Use a calculator to find \( \sqrt[3]{1.09} \approx 1.029327 \).

Step 5: Subtract the Results

Subtract the value of the cube root from the fourth root: \( 1.0297948 - 1.029327 = 0.0004678 \).

Step 6: Verify and Conclude

Review the calculations to check for accuracy. The final result \( 0.0004678 \) seems reasonable given our initial estimates.

Key Concepts

Decimal Approximation

Decimal approximation involves rounding a number to a particular number of decimal places for easier representation or calculations. In our problem, after computing the roots with the calculator, the results are 1.0297948 and 1.029327, respectively. These values give a precise result, but it's beneficial to round for simplicity. If we approximate to four decimal places, 1.0297948 becomes 1.0298 and 1.029327 becomes 1.0293, making it easier to compute manually without significantly losing accuracy. The key here is balancing precision with simplicity, ensuring the results are both manageable and accurate enough for the purpose at hand.

Root Calculation

Root calculation is the process of finding numbers that, when raised to a certain power, equal the original number. In the expression \( \sqrt[4]{1.123} - \sqrt[3]{1.09} \), the first involves finding a number that when raised to the fourth power equals 1.123. Similarly, the second involves finding a number that when cubed equals 1.09. These calculations are crucial in many areas of calculus and can often be simplified using a calculator or approximation if an exact number isn't necessary or possible. Understanding roots is vital as they frequently appear in calculus problems like finding slopes of curves or rates of change.

Mental Estimation

Mental estimation is a handy skill that enables one to quickly predict the outcome of a calculation without the use of calculators. When estimating \( \sqrt[4]{1.123} \) or \( \sqrt[3]{1.09} \), we recognize that both bases (1.123 and 1.09) are slightly greater than 1. Therefore, their roots will also be slightly more than 1. Through quick mental math, approximating them both to roughly 1.03 can immediately alert us that their difference will be very small, possibly around 0, indicating a very minor difference. This skill is useful not just as a sanity check but also in making educated guesses in quantitative reasoning tasks.

Calculator Usage

Using a calculator effectively can save time and improve accuracy. In our case, the roots \( \sqrt[4]{1.123} \) and \( \sqrt[3]{1.09} \) are best found using a scientific calculator's root functions. This type of calculator generally has a root button \((\sqrt{}\)) which lets you input any root power. For these calculations:

• Input 1.123,
• Select 4,
• Use the root function to find the fourth root.

Repeat similarly for 1.09 and the cube root. Calculators provide more digits than necessary, and it's up to us to decide on the sensible level of precision needed for the task. Double-checking the calculator entries and re-evaluating each step are tips to ensure correctness in your results.