Question

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

Short answer

The best decimal approximation of \((3.1415)^{-1/2}\) is about 0.564.

Step-by-step solution

Understand the Expression

The exercise asks us to find the best decimal approximation of the expression \((3.1415)^{-1/2}\). This is equivalent to finding the reciprocal of the square root of 3.1415.

Make a Mental Estimate

Before using a calculator, try to estimate the result. Since 3.1415 is very close to \(\pi\), we know \(\sqrt{\pi}\) is about 1.772. Therefore, \((3.1415)^{-1/2}\) would be approximately \(1/1.772\), which is roughly 0.564. This is a rough mental estimate.

Use Calculator for Precise Value

Input the expression \((3.1415)^{-1/2}\) in your calculator. This can be done by first finding \(\sqrt{3.1415}\) and then taking the reciprocal of that value.

Verify the Calculated Value

Once you have the result from your calculator, ensure it makes sense compared to your mental estimate. The best decimal approximation should be close to the estimate made in Step 2.

Key concepts

Approximation

Approximation is a mathematical technique used to find a close, but not exact, answer to a calculation or problem. It's particularly useful in scenarios where an exact value isn't necessary or where computation resources are limited. Approximation helps simplify complex problems by providing a manageable estimate that is often enough for practical applications.
When approximating, we aim to find a value that is 'good enough' for the purpose at hand. For instance, in the exercise, we need to approximate \((3.1415)^{-1/2}\). This involves estimating its reciprocal square root value. This estimate gives us an idea of what the final calculation might look like before we verify it with a more precise tool, like a calculator.

Square Root

The square root is one of the fundamental operations in mathematics. It helps us find a number that, when multiplied by itself, gives the original number. In our exercise, we work with the square root of 3.1415.
Understanding square roots is crucial because they lay the foundation for more complex operations, such as those encountered in calculus. For example, the square root of a number \( x \) is denoted by \( \sqrt{x} \). Calculating \( \sqrt{3.1415} \) is an essential step in solving the original problem because it helps us to understand what value we must take the reciprocal of in the next step.

Reciprocal

Reciprocals are simply the inverse of a given number. For any non-zero number \( x \), its reciprocal is \( 1/x \). If you multiply a number by its reciprocal, the result is always 1.
In our exercise, after finding the square root of 3.1415, we proceed to determine its reciprocal. This means taking the value of \( \sqrt{3.1415} \) and calculating \( 1/\sqrt{3.1415} \). Reciprocals are a vital part of numerous mathematical procedures, enabling us to solve equations and understand functions more fully.

Estimation

Estimation allows us to quickly gauge the size or nature of a problem without needing an exact answer. It's particularly significant when you need a rapid understanding of potential outcomes or when checking the reasonableness of your findings.
In the step-by-step solution, we made an estimate before using a calculator, looking at how \( \sqrt{3.1415} \) is near the square root of \( \pi \), approximately 1.772. From this, we quickly estimated \( (3.1415)^{-1/2} \) as \( 1/1.772 \), roughly 0.564. Estimation gives us a foundational expectation against which we can compare more precise results obtained later with computational tools.