Problem 36 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{81}{100}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt{\frac{81}{100}}\) is \(\frac{9}{10}\).

Step 1: Identify perfect squares

First, identify that both 81 and 100 are perfect squares. This implies that they both have square roots that are integers.

Step 2: Separate the square root

The square root of a fraction can be separated into the square root of the numerator over the square root of the denominator, using the rule \(\sqrt{a/b} = \sqrt{a}/\sqrt{b}\). So it becomes \(\sqrt{81}\) over \(\sqrt{100}\).

Step 3: Calculate square roots

Now, calculate the square roots of 81 and 100. The square root of 81 is 9 and the square root of 100 is 10.

Step 4: Simplify fraction

Write the expression using the calculated square roots. Our simplified expression becomes 9/10.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, particularly when simplifying expressions. A square root takes a non-negative number and identifies a value which, when multiplied by itself, equals that original number. For example, the square root of 81 is 9 because 9 times 9 equals 81.

• Square roots undo the action of squaring a number.
• They are denoted by the radical symbol \(\sqrt{}\).

Using square roots can simplify complex expressions, especially those involving fractions. By understanding how to extract square roots, students can transform challenging problems into more straightforward expressions.

Perfect Squares

Perfect squares are those numbers that result from squaring an integer. They are fundamental when simplifying square roots because they have integer square roots themselves. For example:

• 9 is a perfect square because it equals \(3^2\).
• 100 is a perfect square because it equals \(10^2\).

Recognizing perfect squares within a problem, like \(\frac{81}{100}\), allows for easy simplification by realizing that \(81\) and \(100\) are \(9^2\) and \(10^2\) respectively. This helps in reducing the complex fraction to simpler terms through their square roots.

Fractions

Fractions represent parts of a whole and are composed of two main components: the numerator and the denominator. In expressions involving square roots, fractions play a crucial role in simplification. When given a fraction under a square root, such as \(\sqrt{\frac{81}{100}}\), important steps are followed:

• Separate the root into individual parts: \(\sqrt{81}\) and \(\sqrt{100}\).
• Simplify each part to turn the fraction into a more workable form.

Focusing on each piece individually can turn an intimidating expression into a simple problem. Ultimately, simplifying fractions under a radical means understanding both concepts inherently.

Numerator and Denominator

In a fraction, the numerator is the number at the top, representing the part of the whole. The denominator is the number at the bottom, showing into how many parts the whole is divided. Simplifying expressions like \(\sqrt{\frac{81}{100}}\) involves dissecting these parts.

• Numerator (81): Simplified as \(\sqrt{81}\ = 9\)
• Denominator (100): Simplified as \(\sqrt{100}\ = 10\)

By addressing the numerator and denominator individually, you can determine their square roots separately, leading to an easier overall expression like \(\frac{9}{10}\). This methodology is critical in breaking down complex fractions.

Integer Square Roots

An integer square root is the positive integer that results from taking the square root of a perfect square. Understanding integer square roots simplifies expressions quickly. For instance:

• The integer square root of 81 is 9.
• The integer square root of 100 is 10.

Integer square roots are significant because they reveal whole number results instantly. In our expression \(\sqrt{\frac{81}{100}}\), recognizing the integer square roots directly simplifies the process, leading to an easier result of \(\frac{9}{10}\). Utilizing integer square roots is key in making swift and accurate calculations in mathematical expressions.