Problem 24 - College Algebra with Modeling and Visualization

Question

If possible, simplify the expression by hand. If you cannot, approximate the answer to the nearest hundredth. Variables represent any real number. $$ \sqrt{121} $$

Step-by-Step Solution

Answer

The simplified expression is 11.

Step 1: Recognize the Expression

Identify the expression as a square root operation. We have \( \sqrt{121} \), which asks for the number that, when multiplied by itself, equals 121.

Step 2: Recall Perfect Squares

Recall that perfect squares are numbers like 1, 4, 9, 16, etc., which are squares of integers. For 121, we look for an integer which, when squared (i.e., multiplied by itself), results in 121.

Step 3: Identify 121 as a Perfect Square

Recognize that 121 is a perfect square because it equals \( 11 \times 11 \). Therefore, \( \sqrt{121} = 11 \).

Step 4: Simplify the Expression

Since 121 is a perfect square, we can directly simplify the expression to 11, which is the integer value of \( \sqrt{121} \).

Key Concepts

Perfect Squares

Perfect squares are numbers that result from squaring an integer. When you square an integer, you multiply it by itself. For example, squaring 3 results in 9, and squaring 5 gives 25. These products are known as perfect squares. Recognizing perfect squares is helpful because they suggest a straightforward solution when dealing with square roots.
For instance, if you encounter a square root of a perfect square, like \(\sqrt{36} \) or \(\sqrt{81} \), you can easily simplify it. 36 and 81 are perfect squares because 36 equals \(6 \times 6\) and 81 equals \(9 \times 9\). Thus, \(\sqrt{36} = 6\) and \(\sqrt{81} = 9\).
Knowing these squares helps in quickly simplifying square root expressions without needing a calculator. Here are few commonly encountered perfect squares:

• 1
• 4
• 9
• 16
• 25
• 36
• 49
• 64
• 81
• 100
• 121

Real Numbers

Real numbers encompass a vast range of numbers that we use in everyday mathematics. This set includes positive numbers, negative numbers, whole numbers, and even fractions and decimals. Basically, if you can put it on a number line, it's a real number.
The realm of real numbers also includes both rational and irrational numbers. Rational numbers are those that can be expressed as fractions, like \(\frac{1}{2}\) or 0.75, while irrational numbers, such as \(\sqrt{2}\) or \(\pi\), cannot be precisely written as simple fractions.
Understanding real numbers is crucial when working with square roots, because the square root of a number, if it exists, is always a real number. So when we calculate \(\sqrt{121}\), which is exactly 11, we see that the simplified square root is also a real number.

Square Root Operation

The square root operation is a fundamental concept in mathematics, used to find a number which, when multiplied by itself, results in the given number. In simpler terms, \(\sqrt{x}\) means "what number times itself equals x?"
When simplifying square roots, you often look for the perfect square component within the number. If the number under the square root is a perfect square, like \(\sqrt{144}\), you can directly find its square root, which is 12. This is because 12 multiplied by 12 gives 144.
Sometimes you need to approximate square roots if they aren't perfect squares. For example, \(\sqrt{50}\) isn't a perfect square, so it requires skills in estimation or using a calculator to get a precise value. However, knowing \(\sqrt{49} = 7\) can help you understand that \(\sqrt{50}\) will be slightly more than 7.
Some tips when working with square roots:

• If it's a perfect square, find the integer that, when squared, gives the original number.
• If not a perfect square, estimate or use a calculator.
• Remember, square roots are defined only for non-negative numbers within real numbers.