Algebra Examples

${(4 + 4 \sqrt{3 i})}^{6}$

Step 1

Use the Binomial Theorem.

$4^{6} + 6 \cdot 4^{5} (4 \sqrt{3 i}) + 15 \cdot 4^{4} {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $4$ to the power of $6$ .

$4096 + 6 \cdot 4^{5} (4 \sqrt{3 i}) + 15 \cdot 4^{4} {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.2

Multiply $4^{5}$ by $4$ by adding the exponents.

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Step 2.1.2.1

Move $4$ .

$4096 + 6 \cdot (4 \cdot 4^{5}) \sqrt{3 i} + 15 \cdot 4^{4} {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.2.2

Multiply $4$ by $4^{5}$ .

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Step 2.1.2.2.1

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

$4096 + 6 \cdot (4^{1} \cdot 4^{5}) \sqrt{3 i} + 15 \cdot 4^{4} {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.2.2.2

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

$4096 + 6 \cdot 4^{1 + 5} \sqrt{3 i} + 15 \cdot 4^{4} {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.2.3

Add $1$ and $5$ .

$4096 + 6 \cdot 4^{6} \sqrt{3 i} + 15 \cdot 4^{4} {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.3

Raise $4$ to the power of $6$ .

$4096 + 6 \cdot 4096 \sqrt{3 i} + 15 \cdot 4^{4} {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.4

Multiply $6$ by $4096$ .

$4096 + 24576 \sqrt{3 i} + 15 \cdot 4^{4} {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.5

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

$4096 + 24576 \sqrt{3 i} + 15 \cdot 256 {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.6

Multiply $15$ by $256$ .

$4096 + 24576 \sqrt{3 i} + 3840 {(4 \sqrt{3 i})}^{2} + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.7

Apply the product rule to $4 \sqrt{3 i}$ .

$4096 + 24576 \sqrt{3 i} + 3840 (4^{2} {\sqrt{3 i}}^{2}) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.8

Raise $4$ to the power of $2$ .

$4096 + 24576 \sqrt{3 i} + 3840 (16 {\sqrt{3 i}}^{2}) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.9

Rewrite ${\sqrt{3 i}}^{2}$ as $3 i$ .

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Step 2.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 i}$ as ${(3 i)}^{\frac{1}{2}}$ .

$4096 + 24576 \sqrt{3 i} + 3840 (16 {({(3 i)}^{\frac{1}{2}})}^{2}) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.9.2

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

$4096 + 24576 \sqrt{3 i} + 3840 (16 {(3 i)}^{\frac{1}{2} \cdot 2}) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.9.3

Combine $\frac{1}{2}$ and $2$ .

$4096 + 24576 \sqrt{3 i} + 3840 (16 {(3 i)}^{\frac{2}{2}}) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

$4096 + 24576 \sqrt{3 i} + 3840 (16 {(3 i)}^{\frac{2}{2}}) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.9.4.2

Rewrite the expression.

$4096 + 24576 \sqrt{3 i} + 3840 (16 {(3 i)}^{1}) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.9.5

Simplify.

$4096 + 24576 \sqrt{3 i} + 3840 (16 (3 i)) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.10

Multiply $3$ by $16$ .

$4096 + 24576 \sqrt{3 i} + 3840 (48 i) + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.11

Multiply $48$ by $3840$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 20 \cdot 4^{3} {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.12

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 20 \cdot 64 {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.13

Multiply $20$ by $64$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 {(4 \sqrt{3 i})}^{3} + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.14

Apply the product rule to $4 \sqrt{3 i}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (4^{3} {\sqrt{3 i}}^{3}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.15

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 {\sqrt{3 i}}^{3}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.16

Rewrite ${\sqrt{3 i}}^{3}$ as $\sqrt{{(3 i)}^{3}}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{{(3 i)}^{3}}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.17

Apply the product rule to $3 i$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{3^{3} i^{3}}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.18

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{27 i^{3}}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.19

Factor out $i^{2}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{27 (i^{2} \cdot i)}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.20

Rewrite $i^{2}$ as $- 1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{27 (- 1 \cdot i)}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.21

Rewrite $- 1 i$ as $- i$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{27 \cdot - 1 i}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.22

Combine exponents.

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Step 2.1.22.1

Factor out negative.

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{- (27 i)}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.22.2

Multiply $27$ by $- 1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{- 27 i}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.23

Rewrite $- 27 i$ as ${(3 i)}^{2} \cdot (3 i)$ .

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Step 2.1.23.1

Factor $- 9$ out of $- 27$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{- 9 (3) i}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.23.2

Rewrite $- 9$ as ${(3 i)}^{2}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{{(3 i)}^{2} \cdot 3 i}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.23.3

Add parentheses.

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 \sqrt{{(3 i)}^{2} \cdot (3 i)}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.24

Pull terms out from under the radical.

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (64 (3 i \sqrt{3 i})) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.25

Multiply $3$ by $64$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 1280 (192 (i \sqrt{3 i})) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.26

Multiply $192$ by $1280$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 (i \sqrt{3 i}) + 15 \cdot 4^{2} {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.27

Raise $4$ to the power of $2$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 15 \cdot 16 {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.28

Multiply $15$ by $16$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 {(4 \sqrt{3 i})}^{4} + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.29

Apply the product rule to $4 \sqrt{3 i}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (4^{4} {\sqrt{3 i}}^{4}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.30

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {\sqrt{3 i}}^{4}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.31

Rewrite ${\sqrt{3 i}}^{4}$ as ${(3 i)}^{2}$ .

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Step 2.1.31.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 i}$ as ${(3 i)}^{\frac{1}{2}}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {({(3 i)}^{\frac{1}{2}})}^{4}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.31.2

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {(3 i)}^{\frac{1}{2} \cdot 4}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.31.3

Combine $\frac{1}{2}$ and $4$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {(3 i)}^{\frac{4}{2}}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.31.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.31.4.1

Factor $2$ out of $4$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {(3 i)}^{\frac{2 \cdot 2}{2}}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.31.4.2

Cancel the common factors.

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Step 2.1.31.4.2.1

Factor $2$ out of $2$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {(3 i)}^{\frac{2 \cdot 2}{2 (1)}}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.31.4.2.2

Cancel the common factor.

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {(3 i)}^{\frac{2 \cdot 2}{2 \cdot 1}}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.31.4.2.3

Rewrite the expression.

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {(3 i)}^{\frac{2}{1}}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.31.4.2.4

Divide $2$ by $1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 {(3 i)}^{2}) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.32

Apply the product rule to $3 i$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 (3^{2} i^{2})) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.33

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 (9 i^{2})) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.34

Rewrite $i^{2}$ as $- 1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 (9 \cdot - 1)) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.35

Multiply $240 (256 (9 \cdot - 1))$ .

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Step 2.1.35.1

Multiply $9$ by $- 1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 (256 \cdot - 9) + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.35.2

Multiply $256$ by $- 9$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 240 \cdot - 2304 + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.35.3

Multiply $240$ by $- 2304$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 6 \cdot 4 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.36

Multiply $6$ by $4$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 {(4 \sqrt{3 i})}^{5} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.37

Apply the product rule to $4 \sqrt{3 i}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (4^{5} {\sqrt{3 i}}^{5}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.38

Raise $4$ to the power of $5$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 {\sqrt{3 i}}^{5}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.39

Rewrite ${\sqrt{3 i}}^{5}$ as $\sqrt{{(3 i)}^{5}}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{{(3 i)}^{5}}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.40

Apply the product rule to $3 i$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{3^{5} i^{5}}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.41

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{243 i^{5}}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.42

Factor out $i^{4}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{243 (i^{4} i)}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.43

Rewrite $i^{4}$ as $1$ .

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Step 2.1.43.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{243 ({(i^{2})}^{2} i)}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.43.2

Rewrite $i^{2}$ as $- 1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{243 ({(- 1)}^{2} i)}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.43.3

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{243 (1 i)}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.44

Multiply $i$ by $1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{243 i}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.45

Rewrite $243 i$ as ${(3^{2})}^{2} \cdot (3 i)$ .

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Step 2.1.45.1

Factor $81$ out of $243$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{81 (3) i}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.45.2

Rewrite $81$ as $9^{2}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{9^{2} \cdot 3 i}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.45.3

Rewrite $9$ as $3^{2}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{{(3^{2})}^{2} \cdot 3 i}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.45.4

Add parentheses.

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 \sqrt{{(3^{2})}^{2} \cdot (3 i)}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.46

Pull terms out from under the radical.

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 (3^{2} \sqrt{3 i})) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.47

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (1024 (9 \sqrt{3 i})) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.48

Multiply $9$ by $1024$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 24 (9216 \sqrt{3 i}) + {(4 \sqrt{3 i})}^{6}$

Step 2.1.49

Multiply $9216$ by $24$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + {(4 \sqrt{3 i})}^{6}$

Step 2.1.50

Apply the product rule to $4 \sqrt{3 i}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4^{6} {\sqrt{3 i}}^{6}$

Step 2.1.51

Raise $4$ to the power of $6$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {\sqrt{3 i}}^{6}$

Step 2.1.52

Rewrite ${\sqrt{3 i}}^{6}$ as ${(3 i)}^{3}$ .

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Step 2.1.52.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 i}$ as ${(3 i)}^{\frac{1}{2}}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {({(3 i)}^{\frac{1}{2}})}^{6}$

Step 2.1.52.2

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {(3 i)}^{\frac{1}{2} \cdot 6}$

Step 2.1.52.3

Combine $\frac{1}{2}$ and $6$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {(3 i)}^{\frac{6}{2}}$

Step 2.1.52.4

Cancel the common factor of $6$ and $2$ .

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Step 2.1.52.4.1

Factor $2$ out of $6$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {(3 i)}^{\frac{2 \cdot 3}{2}}$

Step 2.1.52.4.2

Cancel the common factors.

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Step 2.1.52.4.2.1

Factor $2$ out of $2$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {(3 i)}^{\frac{2 \cdot 3}{2 (1)}}$

Step 2.1.52.4.2.2

Cancel the common factor.

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {(3 i)}^{\frac{2 \cdot 3}{2 \cdot 1}}$

Step 2.1.52.4.2.3

Rewrite the expression.

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {(3 i)}^{\frac{3}{1}}$

Step 2.1.52.4.2.4

Divide $3$ by $1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 {(3 i)}^{3}$

Step 2.1.53

Apply the product rule to $3 i$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 (3^{3} i^{3})$

Step 2.1.54

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

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 (27 i^{3})$

Step 2.1.55

Factor out $i^{2}$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 (27 (i^{2} \cdot i))$

Step 2.1.56

Rewrite $i^{2}$ as $- 1$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 (27 (- 1 \cdot i))$

Step 2.1.57

Rewrite $- 1 i$ as $- i$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 (27 (- i))$

Step 2.1.58

Multiply $- 1$ by $27$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} + 4096 (- 27 i)$

Step 2.1.59

Multiply $- 27$ by $4096$ .

$4096 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} - 552960 + 221184 \sqrt{3 i} - 110592 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $552960$ from $4096$ .

$- 548864 + 24576 \sqrt{3 i} + 184320 i + 245760 i \sqrt{3 i} + 221184 \sqrt{3 i} - 110592 i$

Step 2.2.2

Add $24576 \sqrt{3 i}$ and $221184 \sqrt{3 i}$ .

$- 548864 + 184320 i + 245760 i \sqrt{3 i} + 245760 \sqrt{3 i} - 110592 i$

Step 2.2.3

Subtract $110592 i$ from $184320 i$ .

$- 548864 + 73728 i + 245760 i \sqrt{3 i} + 245760 \sqrt{3 i}$