Simplify $\sqrt{\frac{4}{49}}$.
Answer (1)
Apply the property of square roots to rewrite the expression: 49 4 = 49 4 .
Calculate the square root of the numerator: 4 = 2 .
Calculate the square root of the denominator: 49 = 7 .
Simplify the fraction: 7 2 . The simplified expression is 7 2 .
Explanation
Understanding the Problem We are asked to simplify the square root of a fraction, specifically 49 4 . To do this, we need to understand the properties of square roots and fractions.
Applying the Square Root Property We can use the property that the square root of a fraction is equal to the fraction of the square roots of the numerator and the denominator. In other words, b a = b a . Applying this property to our expression, we get: 49 4 = 49 4
Calculating Square Roots Now, we need to find the square roots of the numerator and the denominator. The square root of 4 is 2, since 2 × 2 = 4 . The square root of 49 is 7, since 7 × 7 = 49 . Therefore, we have: 49 4 = 7 2 .
Final Answer The fraction 7 2 is already in its simplest form, as 2 and 7 have no common factors other than 1. Therefore, the simplified expression is 7 2 .
Examples
Imagine you are tiling a square area and need to determine the length of each tile. If the total area to be tiled is 49 4 square meters, then the side length of the square tile would be 49 4 meters. Simplifying this expression gives you 7 2 meters, which is the length of each tile. This concept is useful in various scaling and measurement problems.
