Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\approx\)s 51-56, find the best decimal approximation that your calculator allows. Begin by making a mental estimate. $$ (\sqrt{3}+1)^{3} $$

Step-by-Step Solution

Answer

The best approximation is 20.445.

Step 1: Estimate the Square Root of 3

Before using the calculator, let's estimate the value of \( \sqrt{3} \). The square root of 3 is approximately 1.732. This is a rough estimation based on knowledge that \( \sqrt{1} = 1 \) and \( \sqrt{4} = 2 \).

Step 2: Add 1 to the Estimated Square Root

Add 1 to our estimated value of \( \sqrt{3} \). This gives us \( 1.732 + 1 = 2.732 \).

Step 3: Calculate the Cube of the Result

Using a calculator, cube the number 2.732. This involves calculating \( 2.732^3 \). Enter 2.732 in the calculator and then raise it to the power of 3. The calculation yields approximately 20.445. This is a more precise answer derived using a calculator.

Key Concepts

Mental Estimation

Mental estimation is a handy tool to quickly guess the approximate result of a mathematical calculation. It's like having a rough sketch before the final masterpiece! Here, we'll explain how to mentally estimate mathematical concepts, focusing specifically on square roots and their computations.

When mentally estimating values like the square root of 3, it's useful to recall easily memorized benchmark values like the square roots of perfect squares. We know that \(\sqrt{1} = 1\) and \(\sqrt{4} = 2\). Therefore, since 3 falls between 1 and 4, the square root of 3 must be somewhere between these two values.

If we need more precision, remember that \(\sqrt{2}\approx 1.414\), so \(\sqrt{3}\approx 1.732\) seems reasonable as a next-step guess. These "mental bookmarks" help us land near the actual number with just brainpower, making mental estimation a quick start before diving into precise calculations with tools like calculators.

Square Root Calculation

Square root calculation is finding a number that, when multiplied by itself, equals the original number. It is often represented with the radical symbol (√). In this context, calculating \(\sqrt{3}\) involves recognizing its approximate decimal value, especially when not working with a calculator.

We start by understanding that 3 is not a perfect square; it doesn't have an integer square root like 4 or 9 do. Using a calculator, we find that \(\sqrt{3} \approx 1.732\). However, being able to estimate it accurately without any tools can be a powerful skill during tests or quick calculations. You can do this by knowing what the square roots of nearby perfect squares are.

Additionally, practicing square root problems helps build intuition around how much larger the square roots of non-perfect squares are from whole numbers. This mental catalog can assist students in navigating through math problems with proficiency.

Cube Computation

In mathematics, cube computation is the process of raising a number to the power of three. This means multiplying the number by itself twice more. For instance, the expression \((\sqrt{3}+1)^3\) requires that we first compute \(\sqrt{3}+1\) and then cube the result.

Consider how we mentally estimated \(\sqrt{3} \approx 1.732\) and then added 1 to get approximately 2.732. The cube of 2.732 is computed as 2.732 multiplied by itself two more times. On a calculator, this precise computation results in 20.445.

To mentally process the cube, break down each multiplication step. Multiply 2.732 by itself to get the square, and then, multiply by 2.732 once again. It's beneficial to try estimating the size of the final number by noting that cubing pushes numbers past their squared values significantly. Even in rough mental computation, it's nice to have expectations such as expecting that the result will be in the 20s, rather than falling short, which aids in verifying the reasonableness of final answers.