5 (sqrt(2) + sqrt(5))
Step-by-step explanation:
Simplify the following:
(5 sqrt(2) + 3 sqrt(5)) (2 sqrt(10) - 5)
Hint: | Multiply 5 sqrt(2) + 3 sqrt(5) and 2 sqrt(10) - 5 together using FOIL.
(5 sqrt(2) + 3 sqrt(5)) (2 sqrt(10) - 5) = (5 sqrt(2)) (-5) + (5 sqrt(2)) (2 sqrt(10)) + (3 sqrt(5)) (-5) + (3 sqrt(5)) (2 sqrt(10)):
-5Γ5 sqrt(2) + 5 sqrt(2)Γ2 sqrt(10) - 5Γ3 sqrt(5) + 3 sqrt(5)Γ2 sqrt(10)
Hint: | Multiply -5 and 5 together.
-5Γ5 = -25:
-25 sqrt(2) + 5 sqrt(2)Γ2 sqrt(10) - 5Γ3 sqrt(5) + 3 sqrt(5)Γ2 sqrt(10)
Hint: | Multiply 5 and 2 together.
5Γ2 = 10:
-25 sqrt(2) + 10 sqrt(2) sqrt(10) - 5Γ3 sqrt(5) + 3 sqrt(5)Γ2 sqrt(10)
Hint: | For a>=0, sqrt(a) sqrt(b) = sqrt(a b). Apply this to sqrt(2) sqrt(10).
sqrt(2) sqrt(10) = sqrt(2Γ10):
-25 sqrt(2) + 10 sqrt(2Γ10) - 5Γ3 sqrt(5) + 3 sqrt(5)Γ2 sqrt(10)
Hint: | Multiply 2 and 10 together.
2Γ10 = 20:
-25 sqrt(2) + 10 sqrt(20) - 5Γ3 sqrt(5) + 3 sqrt(5)Γ2 sqrt(10)
Hint: | Simplify radicals.
sqrt(20) = sqrt(2^2Γ5) = 2 sqrt(5):
-25 sqrt(2) + 10Γ2 sqrt(5) - 5Γ3 sqrt(5) + 3 sqrt(5)Γ2 sqrt(10)
Hint: | Multiply 10 and 2 together.
10Γ2 = 20:
-25 sqrt(2) + 20 sqrt(5) - 5Γ3 sqrt(5) + 3 sqrt(5)Γ2 sqrt(10)
Hint: | Multiply -5 and 3 together.
-5Γ3 = -15:
-25 sqrt(2) + 20 sqrt(5) + -15 sqrt(5) + 3 sqrt(5)Γ2 sqrt(10)
Hint: | Multiply 3 and 2 together.
3Γ2 = 6:
-25 sqrt(2) + 20 sqrt(5) - 15 sqrt(5) + 6 sqrt(5) sqrt(10)
Hint: | For a>=0, sqrt(a) sqrt(b) = sqrt(a b). Apply this to sqrt(5) sqrt(10).
sqrt(5) sqrt(10) = sqrt(5Γ10):
-25 sqrt(2) + 20 sqrt(5) - 15 sqrt(5) + 6 sqrt(5Γ10)
Hint: | Multiply 5 and 10 together.
5Γ10 = 50:
-25 sqrt(2) + 20 sqrt(5) - 15 sqrt(5) + 6 sqrt(50)
Hint: | Simplify radicals.
sqrt(50) = sqrt(2Γ5^2) = 5 sqrt(2):
-25 sqrt(2) + 20 sqrt(5) - 15 sqrt(5) + 6Γ5 sqrt(2)
Hint: | Multiply 6 and 5 together.
6Γ5 = 30:
-25 sqrt(2) + 20 sqrt(5) - 15 sqrt(5) + 30 sqrt(2)
Hint: | Add the numbers in -25 sqrt(2) + 20 sqrt(5) - 15 sqrt(5) + 30 sqrt(2) together.
Add like terms. -25 sqrt(2) + 20 sqrt(5) - 15 sqrt(5) + 30 sqrt(2) = 5 sqrt(2) + 5 sqrt(5):
5 sqrt(2) + 5 sqrt(5)
Hint: | Factor common terms from 5 sqrt(2) + 5 sqrt(5).
Factor 5 out of 5 sqrt(2) + 5 sqrt(5) giving 5 (sqrt(2) + sqrt(5)):
5 (sqrt(2) + sqrt(5))