Problem 20 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \sqrt{\frac{25}{9}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $$\sqrt{\frac{25}{9}}$$ Answer: The simplified expression is $$\frac{5}{3}$$.

Step 1: Identify the numerator and denominator

In the given expression, $$\sqrt{\frac{25}{9}}$$, the numerator is 25 and the denominator is 9.

Step 2: Find the square root of the numerator and denominator separately

Now, find the square root of both 25 and 9. We know that: $$ \sqrt{25} = 5 \\ \sqrt{9} = 3 $$

Step 3: Simplify the given expression

Replace the numerator and the denominator in the given expression with their square roots: $$ \sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} $$ Now, substitute the values found in Step 2: $$ \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3} $$

Step 4: Write the final answer

The simplified expression is: $$ \sqrt{\frac{25}{9}} = \frac{5}{3} $$

Key Concepts

Numerator and Denominator

In any fraction, we can break down the fraction into two main parts: the numerator and the denominator. The numerator is the top part of the fraction while the denominator is the bottom part. Think of the numerator as the 'counting part'—it tells you how many parts you have. On the other hand, the denominator is the 'splitting part'—it indicates into how many parts your whole is divided.
For example, in the fraction \( \frac{25}{9} \), 25 is the numerator and 9 is the denominator. Simply put:

• The numerator (25) shows how many parts we are considering.
• The denominator (9) shows how many equal parts make up a whole.

Understanding the roles of numerator and denominator can simplify the process of solving fraction problems, especially when square roots are involved like in this exercise.

Square Roots

The concept of square roots is deeply rooted in understanding the product of equal factors. A square root of a number is a value that, when multiplied by itself, gives the original number. It’s represented by the symbol \( \sqrt{} \).
For instance, \( \sqrt{25} \) means finding a number which when multiplied by itself equals 25. The answer is 5 because \( 5 \times 5 = 25 \). Similarly, \( \sqrt{9} = 3 \) since \( 3 \times 3 = 9 \).
If you can break a more complex term like \( \sqrt{\frac{25}{9}} \) into individual square roots, it simplifies the calculation. By separating the square root into \( \frac{\sqrt{25}}{\sqrt{9}} \), the problem becomes more manageable and easier to solve.

Fraction Simplification

Simplifying fractions is an essential skill in algebra. It involves reducing fractions to their simplest form. This form is achieved when the numerator and denominator share no common factors other than 1.
When simplifying a fraction like \( \frac{5}{3} \), check if there's a common divisor. Here, 5 and 3 do not share any common factors other than 1, so this is already the simplest form.

• If both numbers in the fraction are derived from perfect squares, like our exercise \( \sqrt{\frac{25}{9}} \), the task of simplification is helped by the properties of square roots.
• A direct calculation of square roots helps us in rearranging and simplifying fractions more efficiently.

Having a fraction in its simplest form makes mathematical operations easier and the expression more elegant.