Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Approximate each expression to the nearest hundredth. $$\sqrt[3]{2.1-6^{2}}$$

Step-by-Step Solution

Answer

-3.24

Step 1: Subtract the Square

First, calculate the expression inside the cube root by subtracting \(6^2\) from 2.1. Compute \(6^2 = 36\), then substitute it back: \(2.1 - 36 = -33.9\).

Step 2: Calculate the Cube Root

Next, find the cube root of \(-33.9\). Use a calculator to approximate \( oot{3}{-33.9} \, ext{or}\, -33.9^{1/3}\), which gives approximately \(-3.24\).

Step 3: Round to Nearest Hundredth

Finally, check if the approximation needs rounding. Since \(-3.24\) already is rounded to the nearest hundredth, we have our approximation.

Key Concepts

Rounding to Nearest Hundredth

Rounding a number to the nearest hundredth is a crucial skill in arithmetic. To do this, identify the decimal place you are rounding to, which in the case of hundredths is the second digit after the decimal point. For example, consider the number - 3.246. The digit in the hundredths place is 4. Look at the digit to its right, which is 6. If this digit is 5 or greater, you round up. - So, 3.246 becomes 3.25. If the digit to the right is less than 5, you simply keep the digit in the hundredths place as it is. - For instance, 3.241 would round to 3.24. This method ensures precision in calculations and simplifies the process when dealing with complex numbers like cube roots.

Simplifying Expressions

Simplifying mathematical expressions helps in breaking down the problem into manageable parts. In our exercise, before we calculate the cube root, it’s essential to simplify the expression within it. - Here, we have the expression \(2.1 - 6^2\).First, compute \(6^2\), which equals 36. Next, substitute this value back into the expression, and subtract it from 2.1.- So, \(2.1 - 36 = -33.9\).The result of this subtraction is the simplified version of the initial expression. It prepares the expression for further mathematical operations like finding a cube root. Keeping expressions simple allows for straightforward application of subsequent operations and minimizes errors.

Calculator Usage

In mathematics, calculators are powerful tools that expedite the process of computation, particularly with complex operations like finding cube roots. In our example, after simplifying the expression to \(-33.9\), a calculator helps determine the cube root.- To find the cube root of \(-33.9\), you can use the cube root function, often denoted as \(\sqrt[3]{x}\), available in scientific calculators.Alternatively, raise the number to the power of \(1/3\), which is equivalent to calculating the cube root:- Enter \(-33.9^{1/3}\) into the calculator.This yields an approximate result of \(-3.24\). Calculators ensure quick, accurate approximations and save you time, especially when rounding numbers to the nearest hundredth. Therefore, they are indispensable for handling expressions involving roots and complex arithmetic.