Question

Вынесите множитель из-под знака корня корень из 680 ,корень из 54/98,корень из 192/245

Answer 1

2корня из 170
3 корня из 3/49
8 корней из 3/245

Answer 2

Simplifying Square Roots

To simplify square roots, we can look for perfect square factors within the radicand (the number under the square root sign) and take them out of the square root. Let's simplify the given expressions:

1. Square root of 680: To simplify the square root of 680, we can look for perfect square factors. The prime factorization of 680 is 2^3 * 5 * 17. We can take out the perfect square factor 2^2 = 4 from 680, which leaves us with 4 * 170 inside the square root. Therefore, the simplified form of the square root of 680 is 2√(170).

2. Square root of 54/98: To simplify the square root of 54/98, we can simplify the fraction first. The greatest common divisor (GCD) of 54 and 98 is 2. Dividing both the numerator and denominator by 2, we get 27/49. Now, let's simplify the square root of 27/49. The prime factorization of 27 is 3^3, and the prime factorization of 49 is 7^2. We can take out the perfect square factor 3 from 27 and the perfect square factor 7 from 49. This leaves us with 3√(3)/7√(1), which simplifies to (3/7)√(3).

3. Square root of 192/245: To simplify the square root of 192/245, let's simplify the fraction first. The GCD of 192 and 245 is 1. Now, let's simplify the square root of 192/245. The prime factorization of 192 is 2^6 * 3, and the prime factorization of 245 is 5 * 7^2. We can take out the perfect square factor 2^2 = 4 from 192 and the perfect square factor 7 from 245. This leaves us with 4√(12)/7√(5). Further simplifying, we have 4√(4 * 3)/7√(5), which becomes 4 * 2√(3)/7√(5). Finally, we get (8/7)√(3/5).

Please let me know if you need any further clarification or assistance!