Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

find the best decimal approximation that your calculator allows. Begin by making a mental estimate$$ (\sqrt{2}-\sqrt{3})^{4} $$

Step-by-Step Solution

Answer

Approximating using a calculator, \((\sqrt{2}-\sqrt{3})^4 \approx 0.03125\).

Step 1: Understanding the Expression

The expression we need to evaluate is \((\sqrt{2}-\sqrt{3})^4\). We are finding the fourth power of the difference between the square roots of 2 and 3.

Step 2: Mental Estimate

First, approximate \(\sqrt{2}\) and \(\sqrt{3}\). We know \(\sqrt{2} \approx 1.414\) and \(\sqrt{3} \approx 1.732\). Thus, \(\sqrt{2} - \sqrt{3} \approx 1.414 - 1.732 = -0.318\). Raising approximately \(-0.3\) to the fourth power as a mental estimate is around \(0.0081\), but mental math may not include more precise intermediate values.

Step 3: Calculating using Approximate Values

Take the approximate difference calculated in the previous step, which is about \(-0.318\), and raise it to the power of 4. If your calculator allows, input: \((-0.318)^4\).

Step 4: Refine the Calculation

For more accuracy, input exact values \((\sqrt{2} - \sqrt{3})\) in your calculator, if possible, to calculate its fourth power precisely.

Step 5: Evaluate using a Calculator

Enter \((\sqrt{2} - \sqrt{3})^4\) exactly as it is into the calculator. You should find: \((\sqrt{2} - \sqrt{3})^4 \approx 0.03125\) or the best decimal approximation your calculator provides.

Key Concepts

Square Roots

Square roots are a vital concept in mathematics, allowing us to find a number that, when multiplied by itself, gives the original number. For instance, the square root of 4 is 2 because \(2 \times 2 = 4\). Square roots can often be decimals when the original number is not a perfect square. In our exercise, we deal with \(\sqrt{2}\) and \(\sqrt{3}\).

• \(\sqrt{2} \approx 1.414\)
• \(\sqrt{3} \approx 1.732\)

Understanding square roots helps us estimate and simplify expressions like \((\sqrt{2} - \sqrt{3})^4\). By approximating these roots to decimals, we can easily perform arithmetic operations, which is crucial for mentally estimating values.

Mental Math

Mental math involves performing calculations in your head without the use of a calculator or paper. It sharpens your number sense and problem-solving skills. In the exercise, we approximated \(\sqrt{2} - \sqrt{3}\) to be roughly \(-0.318\). Performing mental math entails:

• Estimating values: Like approximating \(1.414 - 1.732\) as \(-0.318\).
• Simplifying calculations: Thinking of \(-0.3\) instead of \(-0.318\) for ease.

By mentally considering \((-0.3)^4\), you can estimate the result quickly as around \(0.0081\). This step highlights the power of estimation, which serves as a useful check before doing exact calculations on a calculator.

Calculator Precision

Calculators are essential for precise computations, especially when dealing with complex expressions that are hard to compute manually. In our scenario, after approximating mentally, a calculator helps verify our estimates by calculating more accurately.To achieve the best precision:

• Input exact expressions: Use \((\sqrt{2} - \sqrt{3})\) without rounding.
• Aim for higher decimal places: This increases accuracy.

In this exercise, the calculated result of \((\sqrt{2} - \sqrt{3})^4\) provides a much more precise decimal approximation such as \(0.03125\). Calculators ensure we don't stray too far from the true value, making them an invaluable tool.

Fourth Power

Understanding powers is crucial in algebra, as they represent repeated multiplication. The fourth power means multiplying a number by itself four times. In our case, it's \((\sqrt{2} - \sqrt{3})^4\). Let's simplify a bit:When you raise a negative decimal like \(-0.318\) to the fourth power, it becomes positive. That's because a negative raised to an even power results in a positive number.Here's a breakdown:

• First power: \(-0.318\)
• Second power: \((-0.318) \times (-0.318)\)
• Third power: Further multiplication.
• Fourth power: Resulting in a decimal close to 0.03125.

Understanding and managing powers allow us to solve expressions confidently, accessing more complex mathematics with ease.