Problem 23 - Algebra and Trigonometry

Question

Use the quotient rule to simplify the expressions Assume that \(x>0\). $$ \sqrt{\frac{1}{81}} $$

Step-by-Step Solution

Answer

The simplified expression of \( \sqrt{\frac{1}{81}} \) is \( \frac{1}{9} \).

Step 1: Recognize the number inside the square root

Identify the number inside the square root. Here it is \(\frac{1}{81}\), which is a perfect square, because \(9^2 = 81\).

Step 2: Simplifying square root with perfect square

Since we have \( \frac{1}{81} \) and both 1 and 81 are perfect squares, the square root of \( \frac{1}{81} \) is \( \frac{\sqrt{1}}{\sqrt{81}} \).

Step 3: Perform the Square Root

Square root of 1 is 1 and square root of 81 is 9, therefore the result of \( \frac{\sqrt{1}}{\sqrt{81}} \) is \( \frac{1}{9} \).

Key Concepts

Simplifying Square Roots

Simplifying square roots can often make calculations easier and results more understandable. When we see a square root, like \( \sqrt{\frac{1}{81}} \) in our math problems, simplifying them involves finding the square root of the numerator and the denominator separately. A key point to remember is that the square root of a fraction is the same as the square root of the numerator divided by the square root of the denominator.
You can further simplify the square root of a number by factoring out squares of integers whenever possible. This method reduces the radical expression to its simplest form. In our example, \( \sqrt{1} \) becomes 1 because 1 is the square of itself, and \( \sqrt{81} \) simplifies to 9, as 81 is the square of 9. Ultimately, \( \frac{1}{9} \) is the simplified form of our original square root expression.

Perfect Square

A perfect square is an integer that can be expressed as the product of another integer multiplied by itself. For instance, 16 is a perfect square because it can be written as \( 4 \times 4 \). In the context of simplifying square roots, knowing whether a number is a perfect square can be extremely helpful.
When you identify a perfect square under the square root, you can immediately find its square root without having to perform any additional calculations. This is because the square root of a perfect square is always an integer — a very convenient aspect when dealing with radical expressions. For example, recognizing \( 81 \) in \( \sqrt{\frac{1}{81}} \) as a perfect square lets you quickly conclude that \( \sqrt{81} = 9 \) without the need for guesswork or factorization.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. In our example, \( \sqrt{\frac{1}{81}} \) is a radical expression. These expressions often appear in various mathematical problems and can be challenging to work with until you understand the basic rules and properties of radicals.
For square roots, which are the most common type, remember that only non-negative numbers have real square roots, and the square root of a product is the product of the square roots. Additionally, the square root of a quotient, as seen in our problem, is the quotient of the square roots. Simplifying radical expressions often includes identifying perfect squares within the radicand - the number under the square root symbol - and then using the properties of square roots to simplify the expressions as much as possible.