Problem 45 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest hundredth. $$2 \sqrt{3}$$

Step-by-Step Solution

Answer

The expression \(2 \times \sqrt{3}\) approximates to \(3.46\).

Step 1: Understand the Expression

The expression involves a square root. Specifically, we need to evaluate \(2 \times \sqrt{3}\). First, consider the square root component, \(\sqrt{3}\).

Step 2: Calculate the Square Root

Use a calculator to find the value of \(\sqrt{3}\). The calculator will give you an approximation, which is about \(1.73205\).

Step 3: Multiply by 2

Take the calculated square root and multiply it by 2. So, calculate \(2 \times 1.73205\), which results in approximately \(3.4641\).

Step 4: Round to Nearest Hundredth

Round the result from Step 3 to the nearest hundredth (two decimal places). The number \(3.4641\) rounds to \(3.46\) when approximating to the nearest hundredth.

Key Concepts

Approximations

Approximations play a significant role when dealing with calculations that do not yield exact results.
In the world of mathematics, especially in prealgebra, you may encounter numbers that are irrational, like square roots. These numbers cannot be expressed as a simple fraction and are non-repeating, non-terminating decimals.
Hence, approximations allow us to work with manageable figures without losing significant accuracy.
When approximating, it's crucial to determine how detailed you want your result to be. Rounding numbers to specific decimal places, such as the nearest tenth or hundredth:

• helps in simplifying complex calculations,
• makes results easier to interpret,
• and ensures the tasks remain relevant and easily understandable.

For instance, in our case with the expression involving a square root, approximating to the nearest hundredth makes it efficient for practical use.

Square Roots

The concept of square roots might seem daunting at first, but it becomes easier with practice.
A square root refers to a number which, when multiplied by itself, yields the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9.
However, when dealing with numbers that aren’t perfect squares, like 3, the square root does not result in a neat whole number. Such numbers are irrational, meaning their decimals go on forever without repeating.
For practical applications, like our original exercise, you often need to:

• use a calculator to find an approximate decimal value,
• understand and interpret the resulting number,
• round the number to make it usable, often to two decimal places.

Learning to approximate these values is a fundamental skill that will aid you in more advanced mathematical concepts.

Calculator Usage

Navigating through calculations often requires tools, and calculators are invaluable for this. Knowing how to use a calculator efficiently is an essential skill, especially for solving square roots.
Here's how you can use a calculator to solve an expression involving a square root, like our exercise:

• Firstly, locate and use the square root function key (often represented as √) on your calculator.
• Enter the number you wish to find the square root of, in this case, 3.
• Observe the decimal approximation your calculator provides – around 1.73205 for √3.
• Multiply this approximation by 2, since our expression is 2√3, resulting in a value of approximately 3.4641.
• Finally, round this final result to the nearest hundredth, achieving a clean and precise result of 3.46.

Consistent practice with your calculator builds confidence and ensures accuracy in your mathematical endeavors.