Problem 10 - Algebra 1

Question

Simplify the expression. $$\sqrt{44}$$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{44} \) is \( 2\sqrt{11} \).

Step 1: Identify Perfect Square Factors

Begin by identifying any perfect square factors that 44 might have. A perfect square is the product of a number with itself, and the first few perfect squares are 1, 4, 9, 16, 25, 36, 49, etc. Among these, we see that 4 is a factor of 44.

Step 2: Split the Square Root

Next, split the square root of 44 into the square root of 4 and the square root of 11, because 4 times 11 equals 44. This gives us \( \sqrt{4} \times \sqrt{11} \).

Step 3: Simplify the Perfect Square

Finally, simplify the square root of 4, which is 2. This leaves the expression as \( 2\sqrt{11} \).

Key Concepts

Perfect Square

A perfect square is a number that is the result of squaring a whole number. For example, 4 is a perfect square because it is the product of 2 multiplied by 2. Recognizing perfect squares is helpful when simplifying square roots.

Some commonly seen perfect squares include:

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

Identifying perfect squares within a number allows us to simplify square roots more easily. In our example, we had the number 44. By checking our list, we noticed that 4 is a perfect square that is a factor of 44.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 4 is 2, because 2 \( \times \) 2 equals 4.

In the context of simplifying square roots, understanding how to split a square root into a product of simpler square roots is very useful. For example, if you have \( \sqrt{44} \), you can break it down into \( \sqrt{4} \times \sqrt{11} \). This is possible because 44 can be expressed as a product of 4 and 11.
Recognizing and simplifying square roots through this method reduces more complex expressions into easier forms.

Factorization

Factorization involves breaking down a number into a product of other numbers, or factors, which when multiplied together give the original number. Understanding factorization helps in simplifying square roots by finding perfect square factors.

With the example \( \sqrt{44} \), we looked for factors of 44. Lists of small numbers or tests for divisibility can help. After quick analysis, it's clear that 44 can be broken down into 4, a perfect square, and 11.

• 44 divided by 4 equals 11
• Hence, 44 = 4 \( \times \) 11

By applying factorization, you identify parts of the number that can be simplified separately, making it easier to handle complex square root calculations.