Problem 23 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Define the Problem

We have \( \sqrt{\frac{1}{81}} \) , which is, in essence, equivalent to \( \sqrt{1/81} \) or \( \sqrt{1}/\sqrt{81} \).

Step 2: Applying Square Root on the Numerator and Denominator

The square root of 1 is 1, and the square root of 81 is 9. Thus \( \sqrt{1}/\sqrt{81} \) equates to 1/9.

Step 3: Simplify and Write the Final Answer

Following the steps, we thus simplify \( \sqrt{\frac{1}{81}} \) to 1/9.

Key Concepts

Simplify Expressions

Simplifying expressions is a fundamental skill in mathematics that allows students to reshape mathematical expressions into their simplest form. This process often involves combining like terms, reducing fractions, and employing mathematical rules to make expressions more manageable and easier to understand. For example, when you see a square root over a fraction like \( \sqrt{\frac{1}{81}} \), it may seem complex at first. However, by understanding the properties of square roots and how they interact with fractions, we can simplify this expression in a few straightforward steps.

First, recall that the square root of a fraction can be expressed as the square root of the numerator divided by the square root of the denominator. Using this concept along with the knowledge of basic square roots, such as \( \sqrt{1} = 1 \) and \( \sqrt{81} = 9 \) (since 9 * 9 = 81), we can simplify \( \sqrt{\frac{1}{81}} \) to \( 1/9 \). This is a much cleaner expression and shows the power of simplifying expressions to make them more interpretable.

Square Root

The square root is a fundamental mathematical operation that finds a number, which when multiplied by itself, gives the original number. It is represented by the symbol \( \sqrt{} \) and is critical in simplifying expressions, especially when working with quadratic equations, geometry, and various forms of algebraic problems.

In our exercise \( \sqrt{\frac{1}{81}} \), the square root is applied to a fraction. It is important to remember that \( \sqrt{\frac{a}{b}} \) is equivalent to \( \sqrt{a} / \sqrt{b} \). By applying this property, we can individually find the square root of the numerator and the denominator. The square root of 1 is 1 because 1 * 1 = 1, and the square root of 81 is 9 since 9 * 9 equals 81. When we put this together, we clearly see that the complexity of the square root operation can be broken down into more simple steps resulting in a rational number, which leads us smoothly to our next concept.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction or quotient of two integers, where the numerator is an integer and the denominator is a non-zero integer. They include all the integers, fractions, and decimals that terminate or repeat. In the context of simplifying expressions, converting square roots and other complex mathematical forms into rational numbers can greatly simplify the problem at hand.

In our case, after simplifying \( \sqrt{\frac{1}{81}} \), we arrive at the rational number \( \frac{1}{9} \). This is a clear example of how an initially complicated-looking expression involving a square root results in a simple and elegant rational number. Remember, the power of rational numbers lies in their simplicity and the ease with which we can perform additional arithmetic operations on them as compared to their more complex counterparts.