evaluate sqrt(5+2*(sqrt6)) + sqrt(8-2*(sqrt15))
Use: √(a+2√(ab)+b) = √a + √b
Too: √(a-2√(ab)+b) = √a - √b
√(5+2√6) + √(8-2√15)
= √(2+2√2√3+3) + √(8-2√15)
= √((√2+√3)^2) + √(8-2√15)
= √2 + √3 + √(8-2√15)
= √2 + √3 + √(5-2√3√5+3)
= √2 + √3 + √((√5-√3)^2)
= √2 + √3 + √5 - √3
= √2 + √5
To evaluate the expression sqrt(5+2*(sqrt6)) + sqrt(8-2*(sqrt15)), we first simplify the expression inside each square root individually and then evaluate each square root separately.
1. Simplifying the expression inside the first square root:
5 + 2 * sqrt(6)
There is no further simplification we can do here, so we move on to the next step.
2. Simplifying the expression inside the second square root:
8 - 2 * sqrt(15)
Again, there is no further simplification we can do here, so we move on to the next step.
3. Evaluate each square root separately:
- For the first square root, sqrt(5 + 2 * sqrt(6)):
- Calculate the value inside the square root: 5 + 2 * sqrt(6) = 5 + 2 * 2.449 = 5 + 4.898 ≈ 9.898
- Take the square root of the calculated value: sqrt(9.898) ≈ 3.144
- For the second square root, sqrt(8 - 2 * sqrt(15)):
- Calculate the value inside the square root: 8 - 2 * sqrt(15) = 8 - 2 * 3.873 = 8 - 7.746 ≈ 0.254
- Take the square root of the calculated value: sqrt(0.254) ≈ 0.504
4. Finally, add the evaluated square roots:
sqrt(5 + 2 * sqrt(6)) + sqrt(8 - 2 * sqrt(15))
= 3.144 + 0.504
≈ 3.648
Therefore, the evaluation of sqrt(5+2*(sqrt6)) + sqrt(8-2*(sqrt15)) is approximately 3.648.