Problem 16 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is 36.

Step 1: Evaluate the Square Root

First, identify the square root in the expression \(9 \sqrt{16}\). In this expression, \(\sqrt{16}\) needs to be calculated. The square root of 16 is 4, because 4 multiplied by itself (4 x 4) equals 16. Thus, \(\sqrt{16} = 4\).

Step 2: Simplify the Expression

Now that we know \(\sqrt{16} = 4\), substitute 4 back into the expression. So, the expression becomes \(9 \times 4\).

Step 3: Perform the Multiplication

Finally, multiply 9 by 4 to obtain the simplified result.\(9 \times 4 = 36\).

Key Concepts

Understanding Square Roots

Square roots can be a bit tricky at first, but with a little practice, they become much easier to handle. The square root of a number is what you multiply by itself to get that number. For example, when we talk about the square root of 16, we are asking, "What number times itself equals 16?" The answer is 4, because \(4 \times 4 = 16\). When it's written mathematically, it's shown as \(\sqrt{16} = 4\).

Why are square roots important? They simplify expressions by converting a squared number into its base number. This makes operations like multiplication easier. Remember, not every number has a neat whole number as a square root. However, perfect squares like 16 do, which means their square roots are whole numbers. Perfect squares to remember include 1, 4, 9, 16, 25, and more.

When approaching problems involving square roots, try to identify if the number is a perfect square. If so, find the number that when squared returns the original number.

The Role of Multiplication

Once you have found the square root in an expression, multiplication often comes into play. It's simpler than it sounds! Multiplication is an arithmetic operation where a number is added to itself a certain number of times, reflected in the expression \(a \times b\).

In our solved problem, once we determined that \(\sqrt{16} = 4\), the expression became \(9 \times 4\). This is where multiplication helps us simplify the expression further. Multiplying 9 by 4 gives us 36.

It's helpful to remember some key multiplication facts, which can make these calculations faster and easier.

• Recognize common multiplication pairs, like 5 x 5 = 25 or 6 x 3 = 18.
• Use strategies to break down more complex multiplications, such as the distributive property.

These tips will make handling expressions involving multiplication more manageable.

Evaluating Expressions

Evaluating expressions means determining their value by performing arithmetic operations. It's like solving a puzzle where you put together different pieces, like square roots and multiplication, to find out what the expression equals.

In the original exercise, evaluating the expression started with \(9 \sqrt{16}\). The first step was to find \(\sqrt{16}\), which we calculated to be 4. Then, the simplified expression \(9 \times 4\) was evaluated to find the final value of 36.

To evaluate expressions efficiently:

• Always follow the order of operations: solve any brackets or parentheses first, then exponents like square roots, and follow with multiplication, division, addition, and subtraction (PEMDAS).
• Check if you can simplify any parts of the expression to make calculations easier.

Once you get used to these steps, evaluating expressions becomes much more straightforward and less intimidating.