Problem 74 - A Graphical Approach to College Algebra

Question

Approximate each expression to the nearest hundredth. $$\sqrt[3]{4.5 \times 10^{5}+3.7 \times 10^{2}}$$

Step-by-Step Solution

Answer

The approximate value to the nearest hundredth is 76.53.

Step 1: Simplify Inside the Cube Root

First, simplify the expression inside the cube root: \[ 4.5 \times 10^5 + 3.7 \times 10^2 \]Calculate each term separately. \[ 4.5 \times 10^5 = 450000 \]\[ 3.7 \times 10^2 = 370 \]Add the results: \[ 450000 + 370 = 450370 \]

Step 2: Calculate the Cube Root

Now, find the cube root of the expression obtained:\[ \sqrt[3]{450370} \]Using a calculator, calculate the cube root to several decimal places to ensure accuracy:\[ \sqrt[3]{450370} \approx 76.5258 \]

Step 3: Round to the Nearest Hundredth

Finally, round the result from the previous step to the nearest hundredth. The value 76.5258 rounded to the nearest hundredth is 76.53.

Key Concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a more manageable form. This notation consists of two parts: a number between 1 and 10, and a power of ten. For instance, in our exercise, numbers are expressed as \(4.5 \times 10^5\) and \(3.7 \times 10^2\).

Here's how it works:

• The number before the multiplication sign is called the coefficient. It should be greater than or equal to 1, but less than 10.
• The exponent on the power of ten indicates how many places the decimal point has been moved.

When you calculate the values, you multiply the coefficient by ten raised to the specified power, moving the decimal point accordingly. For \(4.5 \times 10^5\), it simplifies to 450,000 by shifting the decimal 5 places to the right. Meanwhile, \(3.7 \times 10^2\) becomes 370 after moving the decimal 2 places. This is a quick way to manage larger or more intricate numbers.

Step-by-Step Calculation

Breaking down a problem into step-by-step calculations makes it easier to solve complex mathematical tasks. In this particular exercise, the process was approached as follows:

First, start by simplifying within the cube root. Calculate both terms given in scientific notation by converting them to standard numbers:

• Convert \(4.5 \times 10^5\) to 450,000
• Convert \(3.7 \times 10^2\) to 370

Next, add these two results together to get a simplified number to work with inside the cube root, which gives 450,370.

Then, compute the cube root of this sum. Using a calculator aids in accuracy, especially for non-integer roots. The calculation yields \(\sqrt[3]{450370} \approx 76.5258\).

Each step gradually simplifies the expression, leading you closer to the solution, while ensuring accuracy in every calculation.

Rounding to Hundredths

Rounding numbers is essential when you need to express a result to a specific degree of accuracy. "To the nearest hundredth" means you will round the number to two decimal places.

Here is how it is done:

• Look at the third decimal place (thousandth place).
• If the third decimal place is 5 or more, round up the second place by one. Otherwise, keep it the same.

In our solution, the cube root calculated was 76.5258.

Since the third decimal is 5, you round the second digit from 2 to 3, making the rounded result 76.53. This technique ensures that numbers are expressed in a readable format while maintaining precision.