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Chapter 5: Problem 15

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

Short Answer

Expert verified
The expression simplifies to 12.

Step by step solution

01

Identify Perfect Square

First, recognize that both 16 and 9 are perfect squares. This means you can simplify the expression using their square roots.
02

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.
03

Multiply Simplified Roots

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Divide. $$3 \div 4$$

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If you have a penny dated anytime from 1959 through \(1982,\) its original weight was 3.11 grams. If the penny has a date of 1983 or later, the original weight was 2.5 grams. Write the two weights in words.

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