Basic Math Examples

$- 4 t {(3 t^{6} - 1)}^{7}$

Step 1

Use the Binomial Theorem.

$- 4 t ({(3 t^{6})}^{7} + 7 {(3 t^{6})}^{6} \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $3 t^{6}$ .

$- 4 t (3^{7} {(t^{6})}^{7} + 7 {(3 t^{6})}^{6} \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.2

Raise $3$ to the power of $7$ .

$- 4 t (2187 {(t^{6})}^{7} + 7 {(3 t^{6})}^{6} \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.3

Multiply the exponents in ${(t^{6})}^{7}$ .

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Step 2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 t (2187 t^{6 \cdot 7} + 7 {(3 t^{6})}^{6} \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.3.2

Multiply $6$ by $7$ .

$- 4 t (2187 t^{42} + 7 {(3 t^{6})}^{6} \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.4

Apply the product rule to $3 t^{6}$ .

$- 4 t (2187 t^{42} + 7 (3^{6} {(t^{6})}^{6}) \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.5

Raise $3$ to the power of $6$ .

$- 4 t (2187 t^{42} + 7 (729 {(t^{6})}^{6}) \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.6

Multiply the exponents in ${(t^{6})}^{6}$ .

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Step 2.1.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 t (2187 t^{42} + 7 (729 t^{6 \cdot 6}) \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.6.2

Multiply $6$ by $6$ .

$- 4 t (2187 t^{42} + 7 (729 t^{36}) \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.7

Multiply $729$ by $7$ .

$- 4 t (2187 t^{42} + 5103 t^{36} \cdot - 1 + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.8

Multiply $- 1$ by $5103$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 21 {(3 t^{6})}^{5} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.9

Apply the product rule to $3 t^{6}$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 21 (3^{5} {(t^{6})}^{5}) {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.10

Raise $3$ to the power of $5$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 21 (243 {(t^{6})}^{5}) {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.11

Multiply the exponents in ${(t^{6})}^{5}$ .

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Step 2.1.11.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 21 (243 t^{6 \cdot 5}) {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.11.2

Multiply $6$ by $5$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 21 (243 t^{30}) {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.12

Multiply $243$ by $21$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} {(- 1)}^{2} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.13

Raise $- 1$ to the power of $2$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} \cdot 1 + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.14

Multiply $5103$ by $1$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} + 35 {(3 t^{6})}^{4} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.15

Apply the product rule to $3 t^{6}$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} + 35 (3^{4} {(t^{6})}^{4}) {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.16

Raise $3$ to the power of $4$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} + 35 (81 {(t^{6})}^{4}) {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.17

Multiply the exponents in ${(t^{6})}^{4}$ .

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Step 2.1.17.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} + 35 (81 t^{6 \cdot 4}) {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.17.2

Multiply $6$ by $4$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} + 35 (81 t^{24}) {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.18

Multiply $81$ by $35$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} + 2835 t^{24} {(- 1)}^{3} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.19

Raise $- 1$ to the power of $3$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} + 2835 t^{24} \cdot - 1 + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.20

Multiply $- 1$ by $2835$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 35 {(3 t^{6})}^{3} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.21

Apply the product rule to $3 t^{6}$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 35 (3^{3} {(t^{6})}^{3}) {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.22

Raise $3$ to the power of $3$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 35 (27 {(t^{6})}^{3}) {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.23

Multiply the exponents in ${(t^{6})}^{3}$ .

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Step 2.1.23.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 35 (27 t^{6 \cdot 3}) {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.23.2

Multiply $6$ by $3$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 35 (27 t^{18}) {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.24

Multiply $27$ by $35$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} {(- 1)}^{4} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.25

Raise $- 1$ to the power of $4$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} \cdot 1 + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.26

Multiply $945$ by $1$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} + 21 {(3 t^{6})}^{2} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.27

Apply the product rule to $3 t^{6}$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} + 21 (3^{2} {(t^{6})}^{2}) {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.28

Raise $3$ to the power of $2$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} + 21 (9 {(t^{6})}^{2}) {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.29

Multiply the exponents in ${(t^{6})}^{2}$ .

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Step 2.1.29.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} + 21 (9 t^{6 \cdot 2}) {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.29.2

Multiply $6$ by $2$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} + 21 (9 t^{12}) {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.30

Multiply $9$ by $21$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} + 189 t^{12} {(- 1)}^{5} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.31

Raise $- 1$ to the power of $5$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} + 189 t^{12} \cdot - 1 + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.32

Multiply $- 1$ by $189$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} - 189 t^{12} + 7 (3 t^{6}) {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.33

Multiply $3$ by $7$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} - 189 t^{12} + 21 t^{6} {(- 1)}^{6} + {(- 1)}^{7})$

Step 2.1.34

Raise $- 1$ to the power of $6$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} - 189 t^{12} + 21 t^{6} \cdot 1 + {(- 1)}^{7})$

Step 2.1.35

Multiply $21$ by $1$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} - 189 t^{12} + 21 t^{6} + {(- 1)}^{7})$

Step 2.1.36

Raise $- 1$ to the power of $7$ .

$- 4 t (2187 t^{42} - 5103 t^{36} + 5103 t^{30} - 2835 t^{24} + 945 t^{18} - 189 t^{12} + 21 t^{6} - 1)$

Step 2.2

Apply the distributive property.

$- 4 t (2187 t^{42}) - 4 t (- 5103 t^{36}) - 4 t (5103 t^{30}) - 4 t (- 2835 t^{24}) - 4 t (945 t^{18}) - 4 t (- 189 t^{12}) - 4 t (21 t^{6}) - 4 t \cdot - 1$

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

$- 4 \cdot 2187 t \cdot t^{42} - 4 t (- 5103 t^{36}) - 4 t (5103 t^{30}) - 4 t (- 2835 t^{24}) - 4 t (945 t^{18}) - 4 t (- 189 t^{12}) - 4 t (21 t^{6}) - 4 t \cdot - 1$

Step 3.2

Rewrite using the commutative property of multiplication.

$- 4 \cdot 2187 t \cdot t^{42} - 4 \cdot - 5103 t \cdot t^{36} - 4 t (5103 t^{30}) - 4 t (- 2835 t^{24}) - 4 t (945 t^{18}) - 4 t (- 189 t^{12}) - 4 t (21 t^{6}) - 4 t \cdot - 1$

Step 3.3

Rewrite using the commutative property of multiplication.

$- 4 \cdot 2187 t \cdot t^{42} - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 t (- 2835 t^{24}) - 4 t (945 t^{18}) - 4 t (- 189 t^{12}) - 4 t (21 t^{6}) - 4 t \cdot - 1$

Step 3.4

Rewrite using the commutative property of multiplication.

$- 4 \cdot 2187 t \cdot t^{42} - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 t (945 t^{18}) - 4 t (- 189 t^{12}) - 4 t (21 t^{6}) - 4 t \cdot - 1$

Step 3.5

Rewrite using the commutative property of multiplication.

$- 4 \cdot 2187 t \cdot t^{42} - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 t (- 189 t^{12}) - 4 t (21 t^{6}) - 4 t \cdot - 1$

Step 3.6

Rewrite using the commutative property of multiplication.

$- 4 \cdot 2187 t \cdot t^{42} - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 t (21 t^{6}) - 4 t \cdot - 1$

Step 3.7

Rewrite using the commutative property of multiplication.

$- 4 \cdot 2187 t \cdot t^{42} - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} - 4 t \cdot - 1$

Step 3.8

Multiply $- 1$ by $- 4$ .

Step 4

Simplify each term.

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Step 4.1

Multiply $t$ by $t^{42}$ by adding the exponents.

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Step 4.1.1

Move $t^{42}$ .

$- 4 \cdot 2187 (t^{42} t) - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.1.2

Multiply $t^{42}$ by $t$ .

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Step 4.1.2.1

Raise $t$ to the power of $1$ .

$- 4 \cdot 2187 (t^{42} t^{1}) - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 4 \cdot 2187 t^{42 + 1} - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.1.3

Add $42$ and $1$ .

$- 4 \cdot 2187 t^{43} - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.2

Multiply $- 4$ by $2187$ .

$- 8748 t^{43} - 4 \cdot - 5103 t \cdot t^{36} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.3

Multiply $t$ by $t^{36}$ by adding the exponents.

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Step 4.3.1

Move $t^{36}$ .

$- 8748 t^{43} - 4 \cdot - 5103 (t^{36} t) - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.3.2

Multiply $t^{36}$ by $t$ .

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Step 4.3.2.1

Raise $t$ to the power of $1$ .

$- 8748 t^{43} - 4 \cdot - 5103 (t^{36} t^{1}) - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8748 t^{43} - 4 \cdot - 5103 t^{36 + 1} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.3.3

Add $36$ and $1$ .

$- 8748 t^{43} - 4 \cdot - 5103 t^{37} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.4

Multiply $- 4$ by $- 5103$ .

$- 8748 t^{43} + 20412 t^{37} - 4 \cdot 5103 t \cdot t^{30} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.5

Multiply $t$ by $t^{30}$ by adding the exponents.

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Step 4.5.1

Move $t^{30}$ .

$- 8748 t^{43} + 20412 t^{37} - 4 \cdot 5103 (t^{30} t) - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.5.2

Multiply $t^{30}$ by $t$ .

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Step 4.5.2.1

Raise $t$ to the power of $1$ .

$- 8748 t^{43} + 20412 t^{37} - 4 \cdot 5103 (t^{30} t^{1}) - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8748 t^{43} + 20412 t^{37} - 4 \cdot 5103 t^{30 + 1} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.5.3

Add $30$ and $1$ .

$- 8748 t^{43} + 20412 t^{37} - 4 \cdot 5103 t^{31} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.6

Multiply $- 4$ by $5103$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} - 4 \cdot - 2835 t \cdot t^{24} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.7

Multiply $t$ by $t^{24}$ by adding the exponents.

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Step 4.7.1

Move $t^{24}$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} - 4 \cdot - 2835 (t^{24} t) - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.7.2

Multiply $t^{24}$ by $t$ .

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Step 4.7.2.1

Raise $t$ to the power of $1$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} - 4 \cdot - 2835 (t^{24} t^{1}) - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} - 4 \cdot - 2835 t^{24 + 1} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.7.3

Add $24$ and $1$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} - 4 \cdot - 2835 t^{25} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.8

Multiply $- 4$ by $- 2835$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 4 \cdot 945 t \cdot t^{18} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.9

Multiply $t$ by $t^{18}$ by adding the exponents.

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Step 4.9.1

Move $t^{18}$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 4 \cdot 945 (t^{18} t) - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.9.2

Multiply $t^{18}$ by $t$ .

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Step 4.9.2.1

Raise $t$ to the power of $1$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 4 \cdot 945 (t^{18} t^{1}) - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.9.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 4 \cdot 945 t^{18 + 1} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.9.3

Add $18$ and $1$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 4 \cdot 945 t^{19} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.10

Multiply $- 4$ by $945$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} - 4 \cdot - 189 t \cdot t^{12} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.11

Multiply $t$ by $t^{12}$ by adding the exponents.

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Step 4.11.1

Move $t^{12}$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} - 4 \cdot - 189 (t^{12} t) - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.11.2

Multiply $t^{12}$ by $t$ .

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Step 4.11.2.1

Raise $t$ to the power of $1$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} - 4 \cdot - 189 (t^{12} t^{1}) - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} - 4 \cdot - 189 t^{12 + 1} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.11.3

Add $12$ and $1$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} - 4 \cdot - 189 t^{13} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.12

Multiply $- 4$ by $- 189$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} + 756 t^{13} - 4 \cdot 21 t \cdot t^{6} + 4 t$

Step 4.13

Multiply $t$ by $t^{6}$ by adding the exponents.

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Step 4.13.1

Move $t^{6}$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} + 756 t^{13} - 4 \cdot 21 (t^{6} t) + 4 t$

Step 4.13.2

Multiply $t^{6}$ by $t$ .

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Step 4.13.2.1

Raise $t$ to the power of $1$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} + 756 t^{13} - 4 \cdot 21 (t^{6} t^{1}) + 4 t$

Step 4.13.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} + 756 t^{13} - 4 \cdot 21 t^{6 + 1} + 4 t$

Step 4.13.3

Add $6$ and $1$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} + 756 t^{13} - 4 \cdot 21 t^{7} + 4 t$

Step 4.14

Multiply $- 4$ by $21$ .

$- 8748 t^{43} + 20412 t^{37} - 20412 t^{31} + 11340 t^{25} - 3780 t^{19} + 756 t^{13} - 84 t^{7} + 4 t$