Problem 10 - Elementary and Intermediate Algebra

Question

Simplify by taking the roots of the numerator and the denominator. Assume that all variables represent positive numbers. $$ \sqrt{\frac{100}{81}} $$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{10}{9}\).

Step 1 - Express the fraction under the square root

The given problem is \(\frac{\textrm{numerator}}{\textrm{denominator}}\). Express the fraction under the square root: \(\frac{100}{81}\).

Step 2 - Apply the property of square roots to fractions

Use the property \(\frac{\text{numerator}}{\text{denominator}}\) for square roots: \(\frac{\text{numerator}}{\text{denominator}} = \frac{\text{sqrt(numerator)}}{\text{sqrt(denominator)}}\). Apply this property: \[\frac{\textrm{\tiny\textsqrt{100}}}{\textrm{sqrt{81}}}\].

Step 3 - Find the square root of the numerator

The square root of 100 is 10, since \(10^2 = 100\). So, \(\textrm{\tiny\textsqrt{100}} = 10\).

Step 4 - Find the square root of the denominator

The square root of 81 is 9, since \(9^2 = 81\). So, \(\textrm{\tiny\textsqrt{81}} = 9\).

Step 5 - Write the simplified fraction

Combine the results from the previous steps to write the simplified fraction as: \[\frac{10}{9}\].

Key Concepts

Numerator and Denominator

In any fraction, there are two parts, the numerator and the denominator. The numerator is the top number, representing the part of the whole. The denominator is the bottom number, representing the whole or total number of equal parts. For example, in the fraction \(\frac{100}{81}\), 100 is the numerator, and 81 is the denominator. Understanding these components is essential to simplify fractions and apply mathematical operations correctly.

Square Root Properties

Square roots are one of the fundamental concepts in mathematics and understanding their properties is crucial for simplifying expressions. The square root of a number \(x\) is a value \(y\) such that \(y^2 = x\). For example, the square root of 100 is 10, as \(10^2 = 100\), and the square root of 81 is 9, as \(9^2 = 81\). When working with fractions under a square root, we use the property: \(\frac{\text{sqrt}(a)}{\text{sqrt}(b)} = \text{sqrt}(\frac{a}{b})\). This property allows us to separate the square root of a fraction into the square roots of the numerator and the denominator individually.

Fraction Simplification

Simplifying fractions makes them easier to understand and use in further calculations. To simplify, we divide the numerator and the denominator by their greatest common divisor (GCD). In the context of square roots, simplifying fractions involves finding the square root of both the numerator and the denominator, provided the numbers are perfect squares. For instance, given \(\frac{\text{sqrt}(100)}{\text{sqrt}(81)}\), we know that \(10^2 = 100\) and \(9^2 = 81\). Thus, taking the square roots individually, we get \(10\) and \(9\), simplifying the fraction to \(\frac{10}{9}\).