Problem 58 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The expression approximates to 11.312.

Step 1: Identify Like Terms

The given expression is \(3 \sqrt{2} + 5 \sqrt{2}\). First, notice that both terms involve \(\sqrt{2}\), so they are like terms and can be combined.

Step 2: Combine Like Terms

Add the coefficients of the like terms: \(3 + 5 = 8\). This gives the combined expression \(8 \sqrt{2}\).

Step 3: Approximate the Value of \(\sqrt{2}\)

Use a calculator to find the approximate value of \(\sqrt{2}\). This value is approximately \(1.414\).

Step 4: Calculate the Product

Multiply the combined coefficient (8) by the approximate value of \(\sqrt{2}\) (1.414): \(8 \times 1.414 = 11.312\).

Step 5: Write the Final Approximation

Round the product from Step 4 to the nearest thousandth, which is already \(11.312\).

Key Concepts

Understanding Square Roots

Square roots are an essential concept in mathematics, especially when simplifying expressions. A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 times 2 equals 4. The symbol for square root is \(\sqrt{}\).

• Square roots are used to find values that can't be simplified to an exact integer, like \(\sqrt{2}\).
• These values are often irrational numbers, meaning they cannot be expressed as simple fractions.

A calculator is typically used to find decimal approximations, since many roots aren't whole numbers. For instance, \(\sqrt{2}\) is about 1.414. Recognizing these values can help when working with expressions like the one in this exercise: combining terms without losing precision.

Working with Approximations

Approximations are necessary for simplifying expressions that involve irrational numbers. An approximation gives us a nearby value that is easier to work with and understand. This is especially useful when a precise value isn't required, or when we're using a calculator.

• Approximating enables us to handle irrational numbers more conveniently.
• Decimal approximations often round to the nearest thousandth or hundredth, depending on the required precision.

When treating expressions like \(8 \sqrt{2}\), approximating \(\sqrt{2}\) to 1.414 provides a compact way to express and calculate the result. After multiplication, round the resulting number accordingly to ensure it is as accurate as needed for the situation.

Combining Like Terms

Combining like terms is a fundamental process when simplifying algebraic expressions. Like terms are portions of an expression that share the same variables or roots and can be added or subtracted.

• Terms like \(3 \sqrt{2}\) and \(5 \sqrt{2}\) are like terms because they both contain \(\sqrt{2}\).
• To combine them, add their coefficients: \(3 + 5\), resulting in \(8\sqrt{2}\).

Understanding like terms makes it easier to simplify complex expressions by minimizing the number of terms, thereby making calculations more straightforward. This skill not only applies to expressions with roots but also to those involving variables or different powers.