Simplify (- square root of 3+1)^6

(-3+1)6
Use the Binomial Theorem.
(-3)6+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Simplify terms.
Simplify each term.
Apply the product rule to -3.
(-1)636+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Raise -1 to the power of 6.
136+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply 36 by 1.
36+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite 36 as 33.
Use axn=axn to rewrite 3 as 312.
(312)6+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Apply the power rule and multiply exponents, (am)n=amn.
312⋅6+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Combine 12 and 6.
362+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Cancel the common factor of 6 and 2.
Factor 2 out of 6.
32⋅32+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Cancel the common factors.
Factor 2 out of 2.
32⋅32(1)+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Cancel the common factor.
32⋅32⋅1+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite the expression.
331+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Divide 3 by 1.
33+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
33+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
33+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
33+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Raise 3 to the power of 3.
27+6(-3)5⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Apply the product rule to -3.
27+6((-1)535)⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Raise -1 to the power of 5.
27+6(-35)⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite 35 as (35)12.
27+6(-35)⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Raise 3 to the power of 5.
27+6(-243)⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite 243 as 92⋅3.
Factor 81 out of 243.
27+6(-81(3))⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite 81 as 92.
27+6(-92⋅3)⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
27+6(-92⋅3)⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Pull terms out from under the radical.
27+6(-(93))⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply 9 by -1.
27+6(-93)⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply -9 by 6.
27-543⋅1+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply -54 by 1.
27-543+15(-3)4⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Apply the product rule to -3.
27-543+15((-1)434)⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Raise -1 to the power of 4.
27-543+15(134)⋅12+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply 1 by 12 by adding the exponents.
Move 12.
27-543+15(12⋅134)+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply 12 by 1.
Raise 1 to the power of 1.
27-543+15(12⋅1134)+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Use the power rule aman=am+n to combine exponents.
27-543+15(12+134)+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
27-543+15(12+134)+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
27-543+15(1334)+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
27-543+15(1334)+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
One to any power is one.
27-543+15(134)+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply 34 by 1.
27-543+1534+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite 34 as 32.
Use axn=axn to rewrite 3 as 312.
27-543+15(312)4+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Apply the power rule and multiply exponents, (am)n=amn.
27-543+15⋅312⋅4+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Combine 12 and 4.
27-543+15⋅342+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Cancel the common factor of 4 and 2.
Factor 2 out of 4.
27-543+15⋅32⋅22+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Cancel the common factors.
Factor 2 out of 2.
27-543+15⋅32⋅22(1)+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Cancel the common factor.
27-543+15⋅32⋅22⋅1+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite the expression.
27-543+15⋅321+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Divide 2 by 1.
27-543+15⋅32+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
27-543+15⋅32+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
27-543+15⋅32+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
27-543+15⋅32+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Raise 3 to the power of 2.
27-543+15⋅9+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply 15 by 9.
27-543+135+20(-3)3⋅13+15(-3)2⋅14+6(-3)⋅15+16
Apply the product rule to -3.
27-543+135+20((-1)333)⋅13+15(-3)2⋅14+6(-3)⋅15+16
Raise -1 to the power of 3.
27-543+135+20(-33)⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite 33 as (33)12.
27-543+135+20(-33)⋅13+15(-3)2⋅14+6(-3)⋅15+16
Raise 3 to the power of 3.
27-543+135+20(-27)⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite 27 as 32⋅3.
Factor 9 out of 27.
27-543+135+20(-9(3))⋅13+15(-3)2⋅14+6(-3)⋅15+16
Rewrite 9 as 32.
27-543+135+20(-32⋅3)⋅13+15(-3)2⋅14+6(-3)⋅15+16
27-543+135+20(-32⋅3)⋅13+15(-3)2⋅14+6(-3)⋅15+16
Pull terms out from under the radical.
27-543+135+20(-(33))⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply 3 by -1.
27-543+135+20(-33)⋅13+15(-3)2⋅14+6(-3)⋅15+16
Multiply -3 by 20.
27-543+135-603⋅13+15(-3)2⋅14+6(-3)⋅15+16
One to any power is one.
27-543+135-603⋅1+15(-3)2⋅14+6(-3)⋅15+16
Multiply -60 by 1.
27-543+135-603+15(-3)2⋅14+6(-3)⋅15+16
Apply the product rule to -3.
27-543+135-603+15((-1)232)⋅14+6(-3)⋅15+16
Raise -1 to the power of 2.
27-543+135-603+15(132)⋅14+6(-3)⋅15+16
Multiply 1 by 14 by adding the exponents.
Move 14.
27-543+135-603+15(14⋅132)+6(-3)⋅15+16
Multiply 14 by 1.
Raise 1 to the power of 1.
27-543+135-603+15(14⋅1132)+6(-3)⋅15+16
Use the power rule aman=am+n to combine exponents.
27-543+135-603+15(14+132)+6(-3)⋅15+16
27-543+135-603+15(14+132)+6(-3)⋅15+16
27-543+135-603+15(1532)+6(-3)⋅15+16
27-543+135-603+15(1532)+6(-3)⋅15+16
One to any power is one.
27-543+135-603+15(132)+6(-3)⋅15+16
Multiply 32 by 1.
27-543+135-603+1532+6(-3)⋅15+16
Rewrite 32 as 3.
Use axn=axn to rewrite 3 as 312.
27-543+135-603+15(312)2+6(-3)⋅15+16
Apply the power rule and multiply exponents, (am)n=amn.
27-543+135-603+15⋅312⋅2+6(-3)⋅15+16
Combine 12 and 2.
27-543+135-603+15⋅322+6(-3)⋅15+16
Cancel the common factor of 2.
Cancel the common factor.
27-543+135-603+15⋅322+6(-3)⋅15+16
Divide 1 by 1.
27-543+135-603+15⋅31+6(-3)⋅15+16
27-543+135-603+15⋅31+6(-3)⋅15+16
Evaluate the exponent.
27-543+135-603+15⋅3+6(-3)⋅15+16
27-543+135-603+15⋅3+6(-3)⋅15+16
Multiply 15 by 3.
27-543+135-603+45+6(-3)⋅15+16
Multiply -1 by 6.
27-543+135-603+45-63⋅15+16
One to any power is one.
27-543+135-603+45-63⋅1+16
Multiply -6 by 1.
27-543+135-603+45-63+16
One to any power is one.
27-543+135-603+45-63+1
27-543+135-603+45-63+1
162-543-603+45-63+1
207-543-603-63+1
208-543-603-63
208-543-603-63
Subtract 603 from -543.
208-1143-63
Subtract 63 from -1143.
208-1203
208-1203
208-1203
The result can be shown in multiple forms.
Exact Form:
208-1203
Decimal Form:
0.15390309…
Simplify (- square root of 3+1)^6

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