Problem 38 - Algebra 1

Question

Simplify the expression. $$\frac{\sqrt{9}}{\sqrt{49}}$$

Step-by-Step Solution

Answer

The simplified form of \(\frac{\sqrt{9}}{\sqrt{49}}\) is \(\frac{3}{7}\).

Step 1: Understand the square root

When you see \(\sqrt{n}\), consider it as 'what number times itself gives me n'. For instance, \(\sqrt{9}\) means 'what number times itself gives me 9'. The answer is 3, because 3*3 = 9.

Step 2: Apply the square root to the numbers

Let's put the understanding from step 1 into practice:I. Evaluate \(\sqrt{9}\) = 3, because 3*3 = 9.II. Evaluate \(\sqrt{49}\) = 7, because 7*7 = 49.

Step 3: Substitute the square roots

Now substitute the square roots in the original expression with the values we just calculated:\(\frac{\sqrt{9}}{\sqrt{49}} = \frac{3}{7}\)

Key Concepts

Understanding Square Roots

Square roots might seem tricky at first, but they're actually quite straightforward. When you see an expression like \(\sqrt{9}\), it asks what number multiplied by itself gives you the number 9. This is understood as finding that number which, when squared, returns the original value (in this case, 9).
For example:

• \(\sqrt{9}\) equals 3, because \(3 \times 3 = 9\).
• In a similar way, \(\sqrt{49}\) equals 7, because \(7 \times 7 = 49\).

This process is identical for any number. You're simply looking for that hidden value that, when squared, returns the number under the square root symbol \(\sqrt{\ }\). By understanding this concept, you unlock the mystery of simplifying square root expressions.

The Role of Numerator and Denominator

When dealing with fractions, it's crucial to understand the terms 'numerator' and 'denominator'. In any fraction, the numerator is the top number, and it tells how many parts of the whole you're considering. The denominator is the bottom number, which shows the total number of parts in the whole.
For example, in the fraction \(\frac{3}{7}\):

• 3 is the numerator, indicating parts of the whole we're looking at.
• 7 is the denominator, indicating the total number of parts the whole is divided into.

In the original exercise, simplifying \(\frac{\sqrt{9}}{\sqrt{49}}\) involves replacing the square root terms with their simplified values, turning it into a far simpler fraction to work with—\(\frac{3}{7}\). Knowing how numerators and denominators work can simplify handling expressions and help in operations like addition, subtraction, or simplification of fractions.

Simplifying Fractions

Fractions are a way to represent parts of a whole. They can often be simplified to make calculations easier and results clearer. Simplification involves reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1.
For example, take the fraction \(\frac{3}{7}\) derived from \(\frac{\sqrt{9}}{\sqrt{49}}\):

• First, calculate the square roots: \(\sqrt{9} = 3\) and \(\sqrt{49} = 7\).
• Replace the square root terms in the fraction with 3 and 7, respectively, creating \(\frac{3}{7}\).
• Check for common factors in the numerator and denominator. Here, the only common factor is 1, so \(\frac{3}{7}\) is already in its simplest form.

Simplifying fractions is all about making an expression as efficient as possible, paving the way for more straightforward calculations in further mathematics.