Problem 9 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$3 \sqrt{25}$$

Step-by-Step Solution

Answer

The simplified expression is 15.

Step 1: Understand the Square Root

The expression contains the square root of 25, denoted by \( \sqrt{25} \). The square root of a number is a value that, when multiplied by itself, gives the original number. Therefore, the square root of 25 is a number that when squared equals 25.

Step 2: Calculate the Square Root

Determine \( \sqrt{25} \). Since 5 multiplied by itself equals 25 (\(5 \times 5 = 25\)), we have \( \sqrt{25} = 5 \).

Step 3: Multiply by the Coefficient

The original expression is \( 3 \sqrt{25} \). Since we determined that \( \sqrt{25} = 5 \), substitute this value into the expression to get \( 3 \times 5 \).

Step 4: Final Calculation

Multiply the numbers: \( 3 \times 5 = 15 \). Therefore, \( 3 \sqrt{25} = 15 \).

Key Concepts

Understanding Square Roots

A square root is a fundamental concept in math that often perplexes students at first. It involves finding a number that, when multiplied by itself, results in the given number under the square root. For example, consider the square root of 25, written as \( \sqrt{25} \). You are looking for a number that when squared, equals 25.

• \(5 \times 5 = 25\), so the square root of 25 is 5.
• It's like asking, "What number times itself equals 25?" The answer is 5.

Square roots can apply to many contexts, and recognizing these relationships helps solve various types of math problems effectively. Whether you're simplifying expressions or solving equations, understanding the basics of square roots is invaluable.

Working with Coefficients

Coefficients are numerical or constant factors in front of variables or expressions in math. They play a crucial role in calculations, especially when simplifying expressions.

• In the expression \(3 \sqrt{25}\), the number 3 is a coefficient.
• The coefficient multiplies the square root to stretch or scale the magnitude of your result.

When dealing with expressions that involve both coefficients and square roots, first calculate the square root component. Then, simply multiply this result by the coefficient. This approach will help you manage multi-part mathematical operations seamlessly.

Basics of Multiplication

Multiplication is a basic arithmetic operation that involves adding a number to itself a specific number of times. In algebra, it's used extensively to combine terms and simplify expressions. To understand how this works in practice, look at our example:

• The expression is \(3 \times 5\).
• Think of 3 times 5 as adding 5 together three times (5 + 5 + 5), which equals 15.

Multiplication brings together all parts of an expression to give a concise result. Once you've calculated any necessary values (such as a square root), multiplying by the coefficient completes the process. Always check your multiplication steps to ensure you have the correct final result.