Problem 48 - Prealgebra

Question

Compute the exact square root. \(\sqrt{\frac{49}{36}}\)

Step-by-Step Solution

Answer

The exact square root is \( \frac{7}{6} \).

Step 1: Separate the Fraction

Start by expressing the square root of the fraction as a fraction of square roots. That is, \( \sqrt{\frac{49}{36}} = \frac{\sqrt{49}}{\sqrt{36}} \).

Step 2: Calculate the Square Root of the Numerator

The numerator of the fraction is 49, which is a perfect square. Therefore, \( \sqrt{49} = 7 \).

Step 3: Calculate the Square Root of the Denominator

The denominator of the fraction is 36, which is also a perfect square. As such, \( \sqrt{36} = 6 \).

Step 4: Form the Fraction

With the results from the previous two steps, form the fraction for the square root of the original fraction: \( \frac{7}{6} \). This is the exact square root of \( \frac{49}{36} \).

Key Concepts

Perfect Squares

Perfect squares are numbers that are the square of an integer. For example, 49 is a perfect square because it is equal to 7 multiplied by itself: 7 × 7 = 49. Understanding perfect squares is crucial when dealing with square roots, as they help simplify calculations significantly.

• Recognizing perfect squares can speed up your calculations.
• They allow you to quickly pinpoint elements in math problems that will have an exact square root.
• You can always find a perfect square by multiplying any integer by itself. In this exercise, both 49 (numerator) and 36 (denominator) are perfect squares.

The familiarity with a list of small perfect squares, such as 1, 4, 9, 16, 25, 36, 49, and so on, is beneficial. This knowledge allows you to evaluate square roots like in the exercise efficiently. When you spot a perfect square, you can immediately extract its square root without additional calculations.

Fractions

Fractions represent parts of a whole and consist of two components: the numerator and the denominator. In the expression of \[\sqrt{\frac{49}{36}},\]the fraction \(\frac{49}{36}\) has a numerator of 49 and a denominator of 36.

• The fraction line divides these two parts and indicates division.
• A fraction simplifies if both the numerator and the denominator have a common factor.
• Understanding fractions is essential in breaking down more complex mathematical problems into simpler parts, just like separating the square root of each fractional component in the original problem.

Fractions are not only about division but also reflect proportional reasoning in math. Grasping this helps in scenarios where direct multiplication or division simplifies larger problems into more manageable chunks, hence playing a critical role when combining multiple operations, like moving from the square roots of individual numerals to forming a cohesive solution.

Numerator and Denominator Manipulation

Numerator and denominator manipulation are important skills when dealing with fractions, especially when involving operations like square roots. In this exercise, each component of the fraction, 49 and 36, is evaluated separately.

• The numerator (49) is evaluated for its square root independently of the denominator.
• The denominator (36) undergoes the same process; each plays a role in determining the simplified result.
• After finding their roots, the fraction is reassembled into a simplified form.

This approach, transforming \[\sqrt{\frac{49}{36}}\]into \[\frac{\sqrt{49}}{\sqrt{36}}\], is systematic and arises from understanding each component's influence on the fraction as a whole. By breaking down operations and focusing on each part separately, calculations become more manageable. Once calculations are done, they are recombined to find the exact solution. Such manipulations highlight the elegance of math in dissecting and then reconstructing problems for clearer solutions.