Simplify (7 + 2√50) (5 - 2√2). Give your answer in the form a+b√18 where a and b are integers. Show your working clearly.

Question

Simplify (7 + 2√50) (5 - 2√2). Give your answer in the form a+b√18 where a and b are integers. Show your working clearly.

Answer 1

(7 + 2√50) (5 - 2√2)

First, FOIL the expression to expand:

(7)(5)+(7)(-2√2)+(2√50)(5)+(2√50)(-2√2)

Simplify this expression by multiplying:

35-14√2+10√50-4√100

Simplify the terms with radicals and combine like-terms:

35-14√2+10√50-4(10) -----------[√100 = 10]

35-14√2+10√50-40 ---------------[(10)(4) = 40]

-5-14√2+10√50 --------------------[35-40 = -5]

-5-14√2+10√2√25 -----------------[√50 = √2√25]

-5-14√2+10(5)√2 ------------------[√25 = 5]

-5-14√2+50√2 ----------------------[(10)(5) = 50]

-5+36√2 ------------------------------[-14√2+50√2 = 36√2]

Now, we need to convert the expression, -5+36√2, into the form a+b√18, where a and b are integers. We know that √18 = √9√2, and √9 = 3. From this we get the following:

-5+36√2 = -5+(12)(3)√2 ----------------[36 = (12)(3)]

-5+(12)(3)√2 = -5+12√9√2 -------------[3 = √9]

-5+12√9√2 = -5+12√18 -----------------[√9√2 = √18]

Therefore, the final answer is -5+12√18.