Problem 15 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$16 \sqrt{9}$$

Step-by-Step Solution

Answer

The expression simplifies to 12.

Step 1: Identify Perfect Square

First, recognize that both 16 and 9 are perfect squares. This means you can simplify the expression using their square roots.

Step 2: Simplify Perfect Squares

Find the square root of each number. The square root of 16 is 4, and the square root of 9 is 3.

Step 3: Multiply Simplified Roots

Multiply the results from Step 2. Multiply 4 by 3 to get the answer.

Key Concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer multiplied by itself. For example, 16 is a perfect square because it comes from multiplying 4 by 4 (i.e., \(4 \times 4 = 16\)). Recognizing perfect squares is essential when simplifying square roots because it allows you to easily determine the square root without a calculator.

Some common perfect squares include:

• 1 (\(1 \times 1\))
• 4 (\(2 \times 2\))
• 9 (\(3 \times 3\))
• 16 (\(4 \times 4\))
• 25 (\(5 \times 5\))

Recognizing these numbers can help you break down a problem efficiently, such as identifying 9 as a perfect square with a root of 3.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. When you have a number like \(9\), the square root is \(3\) because \(3 \times 3 = 9\). People often simplify expressions involving square roots by noticing if the number under the square root (the radicand) is a perfect square.

For instance, let's say you need to simplify \(\sqrt{16}\). Notice that 16 is a perfect square, as \(4 \times 4 = 16\). Thus, \(\sqrt{16} = 4\). This is why recognizing perfect squares is handy; it lets you quickly find and simplify square roots in expressions.

Multiplying Numbers

Multiplying numbers is a basic arithmetic operation necessary for simplifying square root expressions. After finding the square roots of each perfect square in an expression like \(16 \sqrt{9}\), you multiply the results to arrive at the simplest form.

In the exercise, you simplify \(\sqrt{16}\) to \(4\) and \(\sqrt{9}\) to \(3\). Once you have these square roots, just multiply them together: \(4 \times 3 = 12\). This final step combines your simplified results to give the expression's simplified form, showcasing why multiplying accurately is important.