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express in simplified form

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2 Answers

Hi, Mondrea.
 
to simplify a radical, we break it up into perfect squares. For variables, even exponents are perfect squares.
 
448 = 64 * 7
x8 is a square
y19 = y18 * y
 
So, we have √64 * √7 * √x8 * √y18 * √y
= 8 * √7 * x4 * y9 * √y
= 8x4y9√7y
 
Hope that helps!
Kathye P.
 
BTW, on large numbers, you can also break them down a little at a time, if that helps.
For example, 448 = 4 * 4 * 4 * 7
 
Simplify √(448 x^8 y^19)
 
Find perfect square factors:
 
448 = 4*112 = 4*4*28 = 4*4*4*7
= (4^2)*(2^2)*7 = (8^2)*7
 
x^8 = (x^4)^2
 
y^19 = y*(y^18) = y*(y^9)^2
 
√(448 x^8 y^19)
 
= √( (8^2)*7*(x^4)^2*y*(y^9)^2 )
 
= √( (8^2)*(x^4)^2*(y^9)^2*7*y )
 
= 8*(x^4)*(|y|^9)*√(7 y)
 
The absolute value on y^9 is because √((y^9)^2) is always positive so it's simplified version must be positive too.

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