Question

If 18√8 - 8√18 = √n, what is n? Can you express 18√8 - 8√18 as multiples of the same square root?

Answer 1

Answers:

n = 288

Yes it is possible to express
`18√(8)-8√(18)` as multiples of the same square root (12 multiples of
`√(2)` )

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Step-by-step explanation:

Simplify the first part of the left side

`18√(8) = 18√(4*2)\\\\18√(8) = 18√(4)*√(2)\\\\18√(8) = 18*2*√(2)\\\\18√(8) = 36√(2)`

And do the same for the second part of the left side

`8√(18) = 8√(9*2)\\\\8√(18) = 8√(9)*√(2)\\\\8√(18) = 8*3*√(2)\\\\8√(18) = 24√(2)`

For each simplification, you are trying to factor the stuff under the square root so that you pull out the largest perfect square factor possible.

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The original equation
`18√(8)-8√(18) = √(n)` turns into
`36√(2)-24√(2) = √(n)`

We have the common factor of
`√(2)` so we can combine like terms on the left side ending up with
`12√(2)`

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So,

`18√(8)-8√(18) = √(n)\\\\36√(2)-24√(2) = √(n)\\\\12√(2) = √(n)\\\\√(n) = 12√(2)\\\\\left(√(n)\right)^2 = \left(12√(2)\right)^2\\\\n = 288`

Yes it is possible to express
`18√(8)-8√(18)` as multiples of the same square root. In this case, we can express the left hand side of the original equation as 12 multiples of
`√(2)`

Answer 2

Step-by-step explanation:

squaring both side

(18√8-8√18)^2=(√n)^2

(18√8)^2+(8√18)^2-(2)(8√18)(18√8)=n

after solving it we will get

3744-3456=n

n=288