SOLUTION: simplify the fractional expression: {{{ sqrt(1+(x^3 - 1/4x^3)^2)}}} This is the last question i have in this section an stumped on where to start. Tinking to start/: {{{ s

Algebra ->  Radicals -> SOLUTION: simplify the fractional expression: {{{ sqrt(1+(x^3 - 1/4x^3)^2)}}} This is the last question i have in this section an stumped on where to start. Tinking to start/: {{{ s      Log On


   

Question 544648: simplify the fractional expression:
+sqrt%281%2B%28x%5E3+-+1%2F4x%5E3%29%5E2%29
This is the last question i have in this section an stumped on where to start. Tinking to start/:
+sqrt%281%2B%28x%5E6+-+1%2F4x%5E6%29%29
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then
+sqrt%281%2B%28x%5E6%284x%5E6%29%2F4x%5E6+-+1%2F4x%5E6%29%29
--
next
simplify:
+sqrt%281%2B%284x%5E12+-+1%29%2F4x%5E6%29%29
---
+sqrt%284x%5E6%2F4x%5E6+%2B+%284x%5E12+-+1%29%2F4x%5E6%29%29
--
+sqrt%28%284x%5E6+%2B+4x%5E12-+1%29%29%2Fsqrt%284x%5E6%29%29
---
+sqrt%28%284x%5E6+%2B+4x%5E12-+1%29%29%2F%282%2Aabs%28x%5E3%29%29%29%29%29
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Where do i go from here. Assuming i have done it correctly so far.
Answer by friesr(113) About Me  (Show Source):
You can put this solution on YOUR website!
okay, took me a few tims thru but got the answer. Thanks to JBarnum for getting me thinking of factoring.
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sqrt%28+1+%2B+%28x%5E3+-+1%2F%284x%5E3%29%29%5E2%29
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factor (a - b)^2 = a^2 - 2ab - b^2

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sqrt%281+%2B+x%5E6+-+1%2F2+%2B+1%2F%2816x%5E6%29%29
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combine like terms
sqrt%28x%5E6+%2B+1%2F%2816x%5E2%29+%2B+1%2F2%29
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multiply thru by x^6/x^6 to get rid of the x^6 in the denominator
sqrt%28%28x%5E12+%2B+x%5E6%2F2+%2B+1%2F16%29%281%2Fx%5E6%29%29
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let U = x^6
sqrt%28%28U%5E2+%2B+U%2F2+%2B+1%2F16%29%281%2FU%29%29
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Factor
sqrt%28%28U+%2B+1%2F4%29%5E2+%281%2FU%29%29
-
%28U+%2B+1%2F4%29+%2A+sqrt%281%2FU%29
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bring back in the X^6 in place of U.
%28x%5E6+%2B+1%2F4%29%2A+sqrt%281%2Fx%5E6%29
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%28x%5E6%2B+1%2F4%29%2Fx%5E3
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Multiply thru by 4/4 to get rid of 1/4 fraction
%284x%5E6+%2B+1%29%2F+4x%5E3
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Simplify
%284x%5E6%29%2F%284x%5E3%29+%2B+1%2F%284x%5E3%29
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x%5E3+=+1%2F%284x%5E3%29 Solution