Simplify. [tex]$\sqrt{\frac{36}{81}}$[/tex]
Answer (1)
Express the square root of the fraction as a fraction of square roots: 81 36 = 81 36 .
Evaluate the square roots: 36 = 6 and 81 = 9 , so the expression becomes 9 6 .
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (3): 9 6 = 3 2 .
The simplified expression is 3 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 81 36 . This involves finding the square root of a fraction. Both the numerator and the denominator are perfect squares, which simplifies the process.
Applying the Square Root Property The square root of a fraction can be expressed as a fraction of the square roots: b a = b a . Applying this property to the given expression, we get: 81 36 = 81 36 .
Evaluating Square Roots Now, we evaluate the square roots of the numerator and the denominator. The square root of 36 is 6, and the square root of 81 is 9: 36 = 6 81 = 9 Substituting these values back into the fraction, we have: 81 36 = 9 6 .
Simplifying the Fraction Next, we simplify the fraction 9 6 by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 6 and 9 is 3. Dividing both the numerator and the denominator by 3, we get: 9 6 = 9 ÷ 3 6 ÷ 3 = 3 2 .
Final Answer Therefore, the simplified expression is 3 2 .
Examples
Imagine you are tiling a square area and want to use square tiles. If the area you want to tile is represented by the fraction 81 36 of a larger square, then finding the square root, 3 2 , tells you the side length of the tile you need relative to the larger square. This concept is useful in scaling designs and ensuring proper proportions in construction and design projects.
