Problem 57 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{9}{49}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: \(\sqrt{\frac{9}{49}}\) Answer: \(\frac{3}{7}\)

Step 1: Identify numerator and denominator inside the square root

Our expression is \(\sqrt{\frac{9}{49}}\), in which the numerator inside the square root is \(9\) and the denominator is \(49\).

Step 2: Simplify numerator and denominator

We need to simplify the numerator and denominator by finding the square root of each. The square root of \(9\) is \(3\). The square root of \(49\) is \(7\). So, we have: $$ \sqrt{\frac{9}{49}}=\frac{\sqrt{9}}{\sqrt{49}} $$ Now, we can substitute the square roots: $$ \frac{\sqrt{9}}{\sqrt{49}} = \frac{3}{7} $$

Step 3: Final Answer

The simplified radical expression is: $$ \sqrt{\frac{9}{49}} = \frac{3}{7} $$

Key Concepts

Understanding the Numerator

In the world of fractions and radical expressions, the term **numerator** refers to the top part of a fraction. It indicates how many parts we are considering out of a whole, which is determined by the denominator. For instance, in the fraction \(\frac{9}{49}\), the number \(9\) is the numerator. When dealing with expressions under a square root, like \(\sqrt{\frac{9}{49}}\), the numerator is still the \(9\), but it now plays an important role in the simplification process.

When simplifying radicals, you typically handle the numerator separately from the denominator. In this case, we first find the square root of \(9\), which simplifies to \(3\). It's crucial to separately address the numerator within the context of the radical to help break down the expression into its simplest form.

Exploring the Denominator

The **denominator** is the bottom number in a fraction and it tells you into how many parts the whole is divided. In \(\frac{9}{49}\), the denominator is \(49\). Just like with the numerator, when we are dealing with square roots, especially in expressions such as \(\sqrt{\frac{9}{49}}\), the denominator captures part of the entire radical expression's structure.

Simplifying a radical expression like this involves simplifying the denominator separately. We calculate the square root of \(49\) in this expression. Fortunately, \(49\) is a perfect square, so its square root is \(7\). This calculated value is then used to form the simplified fraction that the expression equals.

The Role of Square Roots

The **square root** is a fundamental concept in mathematics, particularly when simplifying expressions. The square root of a number \(x\) is a number \(y\) such that \(y^2 = x\). In problems involving radicals like \(\sqrt{\frac{9}{49}}\), you will be dealing with the square roots of both the numerator and the denominator.

Each part is simplified individually. In the expression, the square root of \(9\) is \(3\), because \(3 \times 3 = 9\). Similarly, the square root of \(49\) is \(7\), as \(7 \times 7 = 49\). By handling each part separately within the radical, you can simplify the entire expression into a fraction of these two simpler numbers.

Fraction Simplification Simplified

**Fraction simplification** involves reducing a fraction to its simplest form where the numerator and the denominator cannot be any smaller while still being whole numbers. In the context of radicals, especially when dealing with an expression like \(\sqrt{\frac{9}{49}}\), this process begins after computing the individual square roots.

Once you find the square root of the numerator and the denominator, you can rewrite the radical expression as a simple fraction \(\frac{3}{7}\). This fraction is already in its simplest form because \(3\) and \(7\) have no common factors other than \(1\). Thus, simplifying the radical directly translates to simplifying the fractional representation completely.