How do you simplify √(28/343)?

To simplify √(28/343), we use the property that the square root of a quotient equals the quotient of the square roots: √(a/b) = √a/√b. Applying this property: √(28/343) = √28/√343. We've already established that √28 = 2√7. Now we need to simplify √343. Factoring 343: 343 = 7³ = 7² × 7 = 49 × 7. Therefore, √343 = √(49 × 7) = √49 × √7 = 7√7. Putting this together: √(28/343) = (2√7)/(7√7). This expression can be simplified further by canceling the common factor √7 from numerator and denominator: (2√7)/(7√7) = 2/7. This is the fully simplified form.

The simplification process reveals an interesting mathematical pattern. Let's verify our answer by examining the original fraction 28/343 more closely. If we simplify this fraction before taking the square root, we get: 28/343 = (4 × 7)/(49 × 7) = 4/49. Now taking the square root: √(4/49) = √4/√49 = 2/7. This confirms our answer and demonstrates that sometimes it's more efficient to simplify the fraction before applying the radical, especially when both numerator and denominator share common factors that can be canceled.

This problem illustrates several important principles in Class 10 algebra. First, it shows that simplifying radicals in fractions can be approached in multiple ways—you can split the radical into numerator and denominator radicals first, or you can simplify the fraction first and then take the square root. Both methods yield the same result, but depending on the specific numbers involved, one approach might be more efficient than the other. Second, it demonstrates the importance of recognizing common factors: the factor 7 appeared in both the numerator (28 = 4 × 7) and denominator (343 = 49 × 7), allowing for cancellation. Third, it reinforces the concept of rationalizing—in our intermediate step, we had (2√7)/(7√7), which could be viewed as having a radical in the denominator, but canceling √7 from both numerator and denominator gave us the rational number 2/7.