Problem 47 - Prealgebra

Question

Compute the exact square root. \(\sqrt{\frac{144}{25}}\)

Step-by-Step Solution

Answer

The exact value is \(\frac{12}{5}\).

Step 1: Simplify the Expression Inside the Square Root

The expression inside the square root is \(\frac{144}{25}\). Recognize that both 144 and 25 are perfect squares. So, simplify the fraction: \(\sqrt{\frac{144}{25}} = \frac{\sqrt{144}}{\sqrt{25}}\).

Step 2: Compute Each Individual Square Root

Calculate the square root of the numerator and the denominator separately. \(\sqrt{144} = 12\) because \(12 \times 12 = 144\), and \(\sqrt{25} = 5\) because \(5 \times 5 = 25\).

Step 3: Combine the Results

Using the results from the previous step, combine the square roots to find \(\sqrt{\frac{144}{25}} = \frac{12}{5}\). This fraction cannot be simplified further as 12 and 5 have no common factors other than 1.

Key Concepts

Perfect Squares

Perfect squares are numbers that have exact integer square roots. In simpler terms, a perfect square is a number that can be expressed as the square of an integer. For example:

• 144 is a perfect square because it can be written as \(12^2\), where 12 is an integer.
• Similarly, 25 is a perfect square because it can be expressed as \(5^2\).

Recognizing perfect squares is a helpful skill, especially when simplifying square roots like \(\sqrt{144}\) or \(\sqrt{25}\). Knowing these, you can break down square root expressions into simpler forms by taking the square root of each known perfect square. This is particularly useful when dealing with square roots of fractions, as it helps in recognizing which numbers within a fraction have a simple integer identity post squaring.

Simplifying Fractions

Simplifying fractions makes them easier to work with and understand. A fraction like \(\frac{144}{25}\) can often be simplified if there is a common factor between the numerator and the denominator. In the case of perfect squares, the simplification involves taking the square root, reducing the process to simpler terms.

• First, identify any perfect squares in the numerator and the denominator.
• Find their square roots, which in this case are 12 and 5.
• The fraction \(\frac{144}{25}\) simplifies to \(\frac{12}{5}\) because after taking the square roots, the numbers remain whole or in simplest integer form.

Simplification helps to quickly arrive at the most manageable version of a fraction, revealing whether both the square roots are exact integers. If so, as here, it results in a simple fraction with no further possible reduction.

Numerator and Denominator

In a fraction, the numerator and the denominator play key roles. Understanding them is crucial for mathematical operations like simplification and calculation of square roots. The numerator is the top number of a fraction, while the denominator is the lower number that defines the division's scope or size.

• The numerator in \(\frac{144}{25}\) is 144, representing part of a whole.
• The denominator is 25, indicating how many parts make up that whole.

In the context of square roots, each can be handled separately to simplify the operation. Taking the square root of both gives \(\sqrt{144}=12\) for the numerator and \(\sqrt{25}=5\) for the denominator. Comprehending their roles allows for direct simplification without commingling the operations, ensuring clarity in math processes like fraction reduction and square root evaluation.