Problem 69 - Intermediate Algebra

Question

Use a calculator to find each function value. Round to the nearest ten- thousandth. See Example 5 and Using Your Calculator. $$ g(x)=\sqrt[3]{x^{2}+1} $$ a. \(g(6)\) b. \(g(21.57)\)

Step-by-Step Solution

Answer

\( g(6) \approx 3.3322 \); \( g(21.57) \approx 7.7768 \).

Step 1: Identify the Function

We have a function defined as \( g(x) = \sqrt[3]{x^2 + 1} \). For each input value, our goal is to calculate the expression inside the cube root, and then take the cube root of the result.

Step 2: Calculate the Inside of the Root for \( g(6) \)

Substitute \( x = 6 \) into the function to get \( x^2 + 1 = 6^2 + 1 = 36 + 1 = 37 \).

Step 3: Cube Root for \( g(6) \)

Find the cube root of 37 using a calculator. The cube root of 37 is approximately 3.3322 when rounded to the nearest ten-thousandth.

Step 4: Calculate the Inside of the Root for \( g(21.57) \)

Substitute \( x = 21.57 \) into the function to get \( x^2 + 1 = (21.57)^2 + 1 = 465.0249 + 1 = 466.0249 \).

Step 5: Cube Root for \( g(21.57) \)

Find the cube root of 466.0249 using a calculator. The cube root of 466.0249 is approximately 7.7768 when rounded to the nearest ten-thousandth.

Key Concepts

Cube Root

A cube root is a number that, when multiplied by itself twice, equals the original number. It is represented by the radical symbol with a small three on the left, like this: \( \sqrt[3]{x} \). For example, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).

Cube roots are used to "reverse" the process of cubing a number. When you cube a number, you multiply it by itself twice, so the cube root goes backward, finding the original number that was cubed.

When calculating cube roots, you can use a calculator, especially for more complex numbers that don't come out to a whole number. A calculator can quickly return an approximate value for the cube root, especially for non-perfect cube numbers, which can be very useful in solving mathematical problems.

Calculator Usage

Using a calculator can be incredibly helpful when dealing with more complex mathematical functions, like finding the cube root of a number. Most scientific calculators have a function key specifically for cube roots, often found as a dedicated button or within a menu along with other root functions.

To find the cube root of a number on a calculator:

• First, enter the number you want to find the cube root of.
• Next, press the cube root function key or access it through the calculator's function menu.
• The calculator will display the cube root, which you can use in your calculations.

For non-integer cube roots, calculators provide an approximation, crucial for calculations that do not break down neatly into whole numbers. Always ensure your calculator is set to the correct mode for dealing with decimals to ensure accuracy.

Rounding to Nearest Ten-Thousandth

When working with decimals, you may need to round numbers to make them simpler or fit a required degree of accuracy. Rounding to the nearest ten-thousandth involves adjusting a number to four digits after the decimal point. This level of precision is often necessary in scientific calculations and data analysis.

Here's how to round a number to the nearest ten-thousandth:

• Identify the fifth digit after the decimal point.
• If this digit is 5 or greater, round the fourth digit up by one.
• If the fifth digit is less than 5, keep the fourth digit as it is.

For example, to round 3.332176 to the nearest ten-thousandth, you look at the fifth place, which is 7. Since it's 5 or greater, you increase the fourth number (2) by 1, making it 3.3322. This slight adjustment makes your data easier to manage while maintaining a close approximation to the original value.