The number 27+200−−−√−−−−−−−−−√ can be simplified to the form a+b√, where a and b are positive integers. Find the product ab.
To simplify the expression, let's break it down step by step:
1. Start with 27 + 200 - √−−−−−−√. We first need to evaluate the square root.
2. Evaluate the square root of 200. The square root of 200 is approximately 14.142.
3. Substitute the square root with its value: 27 + 14.142.
4. Add 27 and 14.142: 41.142.
So, the simplified form of 27 + 200 - √−−−−−−√ is 41.142. Since we need to express the simplified form in the form a + b√, where a and b are positive integers, the decimal portion of the solution must be written as a square root.
To write 41.142 in the form a + b√, we can approximate it to the nearest whole number as 41 + √0.142.
Now, we have a = 41 and b = 0.142.
To find the product of ab, we multiply a and b:
ab = 41 * 0.142 = 5.822.
Therefore, the product of ab is approximately 5.822.