How do you solve (5√2+3√5) (2√10-5)

Question

Sofia Velasquez

Mathematics

21 January, 19:06

How do you solve (5√2+3√5) (2√10-5)

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Answers (1)

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Caitlin · Answer 1

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)):

-5*5 sqrt (2) + 5 sqrt (2) * 2 sqrt (10) - 5*3 sqrt (5) + 3 sqrt (5) * 2 sqrt (10)

Hint: | Multiply - 5 and 5 together.

-5*5 = - 25:

-25 sqrt (2) + 5 sqrt (2) * 2 sqrt (10) - 5*3 sqrt (5) + 3 sqrt (5) * 2 sqrt (10)

Hint: | Multiply 5 and 2 together.

5*2 = 10:

-25 sqrt (2) + 10 sqrt (2) sqrt (10) - 5*3 sqrt (5) + 3 sqrt (5) * 2 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 (2*10):

-25 sqrt (2) + 10 sqrt (2*10) - 5*3 sqrt (5) + 3 sqrt (5) * 2 sqrt (10)

Hint: | Multiply 2 and 10 together.

2*10 = 20:

-25 sqrt (2) + 10 sqrt (20) - 5*3 sqrt (5) + 3 sqrt (5) * 2 sqrt (10)

Hint: | Simplify radicals.

sqrt (20) = sqrt (2^2*5) = 2 sqrt (5):

-25 sqrt (2) + 10*2 sqrt (5) - 5*3 sqrt (5) + 3 sqrt (5) * 2 sqrt (10)

Hint: | Multiply 10 and 2 together.

10*2 = 20:

-25 sqrt (2) + 20 sqrt (5) - 5*3 sqrt (5) + 3 sqrt (5) * 2 sqrt (10)

Hint: | Multiply - 5 and 3 together.

-5*3 = - 15:

-25 sqrt (2) + 20 sqrt (5) + - 15 sqrt (5) + 3 sqrt (5) * 2 sqrt (10)

Hint: | Multiply 3 and 2 together.

3*2 = 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 (5*10):

-25 sqrt (2) + 20 sqrt (5) - 15 sqrt (5) + 6 sqrt (5*10)

Hint: | Multiply 5 and 10 together.

5*10 = 50:

-25 sqrt (2) + 20 sqrt (5) - 15 sqrt (5) + 6 sqrt (50)

Hint: | Simplify radicals.

sqrt (50) = sqrt (2*5^2) = 5 sqrt (2):

-25 sqrt (2) + 20 sqrt (5) - 15 sqrt (5) + 6*5 sqrt (2)

Hint: | Multiply 6 and 5 together.

6*5 = 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)):

Answer: 5 (sqrt (2) + sqrt (5))