Problem 57 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest thousandth. $$2 \sqrt{3}+5 \sqrt{3}$$

Step-by-Step Solution

Answer

The expression approximates to 12.124.

Step 1: Combine Like Terms

Recognize that both terms in the expression \(2 \sqrt{3} + 5 \sqrt{3}\) are like terms because they both involve \(\sqrt{3}\). Add the coefficients of the like terms together: \(2 + 5 = 7\). Therefore, the expression simplifies to \(7 \sqrt{3}\).

Step 2: Calculate the Square Root

Use a calculator to find the square root of 3. Enter \( \sqrt{3} \) into the calculator to obtain approximately \(1.732\).

Step 3: Multiply the Result

Multiply the simplified coefficient \(7\) by the approximate value of \(\sqrt{3}\), which is \(1.732\). Calculate \(7 \times 1.732\) to get approximately \(12.124\).

Step 4: Round to the Nearest Thousandth

Look at the result from Step 3, \(12.124\), and ensure it is rounded to the nearest thousandth. Since the number already has three decimal places, no further rounding is needed.

Key Concepts

Approximating Expressions

Approximating expressions involves rounding a decimal or a result to a specified degree of precision. This is crucial when dealing with complex numbers or irrational numbers, like square roots, especially when a precise result is less essential. In this context, understanding how to approximate ensures results are manageable and comprehensible.
When required to approximate, it's usually to a certain place value such as the thousandth, which is three places after the decimal. For instance, to approximate 12.12345 to the nearest thousandth:

• Look at the fourth decimal digit: if it's 5 or higher, round up.
• If it's 4 or lower, round down.

Therefore, 12.12345 becomes 12.123. This rounding technique is common for bringing complex calculator results into everyday use.

Combining Like Terms

Combining like terms is fundamental in algebra for simplifying expressions and making calculations more straightforward. Like terms are terms in an algebraic expression that have the same variable raised to the same power. For instance, in the expression \(2 \sqrt{3} + 5 \sqrt{3}\), both terms are 'like terms' because they share \(\sqrt{3}\).
To combine them, simply add the coefficients (the numbers in front). Here, you would add 2 and 5 to get 7. The expression then simplifies to \(7 \sqrt{3}\). This simplification reduces the complexity of further calculations and makes it easier to estimate the final result when using a calculator.

Square Root Calculation

Calculating the square root of a number is fundamental in various branches of mathematics and applies to many real-life scenarios. When a calculator is used, it's often to find the square root of numbers that don't have perfect integer square roots, like 3 in our example.
To calculate \(\sqrt{3}\) using a calculator:

• Enter the number 3.
• Press the square root function button, often labeled as \(\sqrt{}\) or "sqrt".
• The calculator will display the approximate square root, which is around 1.732 (to three decimal places).

This function is vital for tackling problems involving quadratic equations, geometry calculations, and whenever finding lengths of sides in right-angled triangles. Accurate calculator use and understanding result approximation will significantly enhance your confidence in handling square roots.