# Simplify (2t+1/t)^7

(2t+1t)7
Use the Binomial Theorem.
(2t)7+7(2t)61t+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Simplify each term.
Apply the product rule to 2t.
27t7+7(2t)61t+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Raise 2 to the power of 7.
128t7+7(2t)61t+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 2t.
128t7+7(26t6)1t+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Cancel the common factor of t.
Factor t out of 7(26t6).
128t7+t(7(26t5))1t+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Cancel the common factor.
128t7+t(7(26t5))1t+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Rewrite the expression.
128t7+7(26t5)+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
128t7+7(26t5)+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Raise 2 to the power of 6.
128t7+7(64t5)+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Multiply 64 by 7.
128t7+448t5+21(2t)5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 2t.
128t7+448t5+21(25t5)(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Raise 2 to the power of 5.
128t7+448t5+21(32t5)(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Multiply 32 by 21.
128t7+448t5+672t5(1t)2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 1t.
128t7+448t5+672t512t2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Cancel the common factor of t2.
Factor t2 out of 672t5.
128t7+448t5+t2(672t3)12t2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Cancel the common factor.
128t7+448t5+t2(672t3)12t2+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Rewrite the expression.
128t7+448t5+672t3⋅12+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
128t7+448t5+672t3⋅12+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
One to any power is one.
128t7+448t5+672t3⋅1+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Multiply 672 by 1.
128t7+448t5+672t3+35(2t)4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 2t.
128t7+448t5+672t3+35(24t4)(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Raise 2 to the power of 4.
128t7+448t5+672t3+35(16t4)(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Multiply 16 by 35.
128t7+448t5+672t3+560t4(1t)3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 1t.
128t7+448t5+672t3+560t413t3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Cancel the common factor of t3.
Factor t3 out of 560t4.
128t7+448t5+672t3+t3(560t)13t3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Cancel the common factor.
128t7+448t5+672t3+t3(560t)13t3+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Rewrite the expression.
128t7+448t5+672t3+560t⋅13+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
128t7+448t5+672t3+560t⋅13+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
One to any power is one.
128t7+448t5+672t3+560t⋅1+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Multiply 560 by 1.
128t7+448t5+672t3+560t+35(2t)3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 2t.
128t7+448t5+672t3+560t+35(23t3)(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Raise 2 to the power of 3.
128t7+448t5+672t3+560t+35(8t3)(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Multiply 8 by 35.
128t7+448t5+672t3+560t+280t3(1t)4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 1t.
128t7+448t5+672t3+560t+280t314t4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Cancel the common factor of t3.
Factor t3 out of 280t3.
128t7+448t5+672t3+560t+t3⋅28014t4+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Factor t3 out of t4.
128t7+448t5+672t3+560t+t3⋅28014t3t+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Cancel the common factor.
128t7+448t5+672t3+560t+t3⋅28014t3t+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Rewrite the expression.
128t7+448t5+672t3+560t+28014t+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
128t7+448t5+672t3+560t+28014t+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Combine 280 and 14t.
128t7+448t5+672t3+560t+280⋅14t+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
One to any power is one.
128t7+448t5+672t3+560t+280⋅1t+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Multiply 280 by 1.
128t7+448t5+672t3+560t+280t+21(2t)2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 2t.
128t7+448t5+672t3+560t+280t+21(22t2)(1t)5+7(2t)(1t)6+(1t)7
Raise 2 to the power of 2.
128t7+448t5+672t3+560t+280t+21(4t2)(1t)5+7(2t)(1t)6+(1t)7
Multiply 4 by 21.
128t7+448t5+672t3+560t+280t+84t2(1t)5+7(2t)(1t)6+(1t)7
Apply the product rule to 1t.
128t7+448t5+672t3+560t+280t+84t215t5+7(2t)(1t)6+(1t)7
Cancel the common factor of t2.
Factor t2 out of 84t2.
128t7+448t5+672t3+560t+280t+t2⋅8415t5+7(2t)(1t)6+(1t)7
Factor t2 out of t5.
128t7+448t5+672t3+560t+280t+t2⋅8415t2t3+7(2t)(1t)6+(1t)7
Cancel the common factor.
128t7+448t5+672t3+560t+280t+t2⋅8415t2t3+7(2t)(1t)6+(1t)7
Rewrite the expression.
128t7+448t5+672t3+560t+280t+8415t3+7(2t)(1t)6+(1t)7
128t7+448t5+672t3+560t+280t+8415t3+7(2t)(1t)6+(1t)7
Combine 84 and 15t3.
128t7+448t5+672t3+560t+280t+84⋅15t3+7(2t)(1t)6+(1t)7
One to any power is one.
128t7+448t5+672t3+560t+280t+84⋅1t3+7(2t)(1t)6+(1t)7
Multiply 84 by 1.
128t7+448t5+672t3+560t+280t+84t3+7(2t)(1t)6+(1t)7
Multiply 2 by 7.
128t7+448t5+672t3+560t+280t+84t3+14t(1t)6+(1t)7
Apply the product rule to 1t.
128t7+448t5+672t3+560t+280t+84t3+14t16t6+(1t)7
Cancel the common factor of t.
Factor t out of 14t.
128t7+448t5+672t3+560t+280t+84t3+t⋅1416t6+(1t)7
Factor t out of t6.
128t7+448t5+672t3+560t+280t+84t3+t⋅1416t⋅t5+(1t)7
Cancel the common factor.
128t7+448t5+672t3+560t+280t+84t3+t⋅1416t⋅t5+(1t)7
Rewrite the expression.
128t7+448t5+672t3+560t+280t+84t3+1416t5+(1t)7
128t7+448t5+672t3+560t+280t+84t3+1416t5+(1t)7
Combine 14 and 16t5.
128t7+448t5+672t3+560t+280t+84t3+14⋅16t5+(1t)7
One to any power is one.
128t7+448t5+672t3+560t+280t+84t3+14⋅1t5+(1t)7
Multiply 14 by 1.
128t7+448t5+672t3+560t+280t+84t3+14t5+(1t)7
Apply the product rule to 1t.
128t7+448t5+672t3+560t+280t+84t3+14t5+17t7
One to any power is one.
128t7+448t5+672t3+560t+280t+84t3+14t5+1t7
128t7+448t5+672t3+560t+280t+84t3+14t5+1t7
Simplify (2t+1/t)^7

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