Add 31−−√+31−−√.
To add the expressions 31-√31-√, we simply combine like terms.
31-√+31-√ can be written as (31-√31-√).
Since both terms have the same radical (√31), we can combine them.
(31-√31-√) = 31 + 31 - 2√31
Therefore, 31-√+31-√ can be simplified to 62 - 2√31.
To add two square roots together, you can first simplify each square root, and then add them together.
Let's simplify each square root separately:
√31 can be simplified as follows:
Start by factoring 31 into its prime factors: 31 is a prime number, so it cannot be factored.
Since there are no perfect square factors, we cannot simplify √31 any further.
Now let's simplify the second square root, √31:
Again, we start by factoring 31: 31 is already a prime number.
Now, we can add the simplified square roots together:
√31 + √31 = 2√31
Therefore, the sum of √31 + √31 is 2√31.
To add √31 + √31, we first simplify each radical separately.
Step 1: Simplify √31
To simplify the square root of 31, we look for the largest perfect square that divides evenly into 31. The largest perfect square less than 31 is 25 (5 * 5).
Since 25 is less than 31, we can split 31 into two parts: 25 and 6.
√31 = √(25 + 6) = √25 + √6 = 5√1 + √6 = 5 + √6
Step 2: Add the simplified radicals
Now that we have √31 simplified as 5 + √6, we can add the two radicals together:
(5 + √6) + (5 + √6) = 10 + 2√6
Therefore, the simplified expression is 10 + 2√6.