Problem 42 - Prealgebra

Question

Compute the exact square root. \(\sqrt{\frac{625}{324}}\)

Step-by-Step Solution

Answer

The exact square root is \( \frac{25}{18} \).

Step 1: Understand the Problem

We need to find the exact square root of \( \frac{625}{324} \). This means we need to find a number that, when squared, gives \( \frac{625}{324} \).

Step 2: Simplify the Expression

Find the square root of the numerator \( 625 \) and the square root of the denominator \( 324 \) separately. This will help us simplify \( \sqrt{\frac{625}{324}} \) into \( \frac{\sqrt{625}}{\sqrt{324}} \).

Step 3: Calculate the Square Roots

The square root of 625 is 25, since \( 25 \times 25 = 625 \). The square root of 324 is 18, since \( 18 \times 18 = 324 \).

Step 4: Divide the Simplified Roots

Now, divide the square root of the numerator by the square root of the denominator: \( \frac{25}{18} \).

Step 5: Confirm the Result

To ensure the result is correct, square the result: \( \left(\frac{25}{18}\right)^2 = \frac{625}{324} \), which confirms our computation is accurate.

Key Concepts

Understanding Numerator and Denominator

When dealing with fractions, it's essential to know the parts that make up a fraction. A fraction consists of two parts: the numerator and the denominator. These are the numbers that appear above and below the fraction line, respectively.
The **numerator** is the top number in a fraction. It shows how many parts of the whole or group are being considered. For instance, in the fraction \( \frac{625}{324} \), the number 625 is the numerator. It indicates the specific count of parts we are focusing on.
On the other hand, the **denominator** is the bottom number. It reveals how many equal parts the whole is divided into. Using our example, 324 is the denominator, telling us that the whole is split into 324 equal segments. Remember, understanding these terms is crucial for any operations on fractions, be it addition, multiplication, or finding square roots.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible while keeping its value unchanged. This often means reducing the fraction to its smallest terms. But why simplify? Well, simpler fractions are easier to work with and understand.
To simplify a fraction, we look for the greatest common divisor (GCD) of the numerator and the denominator. By dividing both parts by their GCD, we can reduce the fraction. However, when finding the square root, like in our exercise \( \sqrt{\frac{625}{324}} \), simplifying becomes more about evaluating each part separately.

• First, identify the square root of each part. For 625, it's 25, because \( 25 \times 25 = 625 \).
• Next, do the same for 324 to find it's 18, as \( 18 \times 18 = 324 \).

Once you find these roots, rewrite the original problem as \( \frac{25}{18} \), which is not exactly reducing in the traditional sense but is akin to simplification in the context of square roots. This fraction \( \frac{25}{18} \) is the simplest form of the square root expression because you can't simplify it further under whole numbers.

Exact Computation in Square Roots

Exact computation involves calculating without approximations. When it comes to square roots, often people opt for decimal approximations for ease. However, this approach can lack precision, especially in fields requiring high accuracy.
In our exercise \( \sqrt{\frac{625}{324}} \), exact computation ensures we find the precise square root, represented completely as \( \frac{25}{18} \). This approach is hugely beneficial because:

• It avoids rounding errors that can occur with decimals.
• It keeps calculations neat and precise, ensuring no loss of information.
• In academic and scientific settings, exact values are not just helpful but sometimes compulsory for accuracy.

By doing exact computation, as we did here, you verify that the square of your result truly is the original fraction \( \frac{625}{324} \). This is because \( \left(\frac{25}{18}\right)^2 = \frac{625}{324} \), proving that staying exact pays off in keeping math precise and dependable.