Precalculus Examples

${(\sqrt{5} - 2 i)}^{5}$

Step 1

Use the Binomial Theorem.

${\sqrt{5}}^{5} + 5 {\sqrt{5}}^{4} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${\sqrt{5}}^{5}$ as $\sqrt{5^{5}}$ .

$\sqrt{5^{5}} + 5 {\sqrt{5}}^{4} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.2

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

$\sqrt{3125} + 5 {\sqrt{5}}^{4} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.3

Rewrite $3125$ as $25^{2} \cdot 5$ .

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Step 2.1.3.1

Factor $625$ out of $3125$ .

$\sqrt{625 (5)} + 5 {\sqrt{5}}^{4} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.3.2

Rewrite $625$ as $25^{2}$ .

$\sqrt{25^{2} \cdot 5} + 5 {\sqrt{5}}^{4} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.4

Pull terms out from under the radical.

$25 \sqrt{5} + 5 {\sqrt{5}}^{4} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.5

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$25 \sqrt{5} + 5 {(5^{\frac{1}{2}})}^{4} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.5.2

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

$25 \sqrt{5} + 5 \cdot 5^{\frac{1}{2} \cdot 4} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.5.3

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

$25 \sqrt{5} + 5 \cdot 5^{\frac{4}{2}} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.5.4

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

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Step 2.1.5.4.1

Factor $2$ out of $4$ .

$25 \sqrt{5} + 5 \cdot 5^{\frac{2 \cdot 2}{2}} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.5.4.2

Cancel the common factors.

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Step 2.1.5.4.2.1

Factor $2$ out of $2$ .

$25 \sqrt{5} + 5 \cdot 5^{\frac{2 \cdot 2}{2 (1)}} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.5.4.2.2

Cancel the common factor.

$25 \sqrt{5} + 5 \cdot 5^{\frac{2 \cdot 2}{2 \cdot 1}} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.5.4.2.3

Rewrite the expression.

$25 \sqrt{5} + 5 \cdot 5^{\frac{2}{1}} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.5.4.2.4

Divide $2$ by $1$ .

$25 \sqrt{5} + 5 \cdot 5^{2} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.6

Multiply $5$ by $5^{2}$ by adding the exponents.

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Step 2.1.6.1

Multiply $5$ by $5^{2}$ .

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Step 2.1.6.1.1

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

$25 \sqrt{5} + 5^{1} \cdot 5^{2} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.6.1.2

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

$25 \sqrt{5} + 5^{1 + 2} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.6.2

Add $1$ and $2$ .

$25 \sqrt{5} + 5^{3} (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.7

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

$25 \sqrt{5} + 125 (- 2 i) + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.8

Multiply $- 2$ by $125$ .

$25 \sqrt{5} - 250 i + 10 {\sqrt{5}}^{3} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.9

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$25 \sqrt{5} - 250 i + 10 \sqrt{5^{3}} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.10

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

$25 \sqrt{5} - 250 i + 10 \sqrt{125} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.11

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 2.1.11.1

Factor $25$ out of $125$ .

$25 \sqrt{5} - 250 i + 10 \sqrt{25 (5)} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.11.2

Rewrite $25$ as $5^{2}$ .

$25 \sqrt{5} - 250 i + 10 \sqrt{5^{2} \cdot 5} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.12

Pull terms out from under the radical.

$25 \sqrt{5} - 250 i + 10 (5 \sqrt{5}) {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.13

Multiply $5$ by $10$ .

$25 \sqrt{5} - 250 i + 50 \sqrt{5} {(- 2 i)}^{2} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.14

Apply the product rule to $- 2 i$ .

$25 \sqrt{5} - 250 i + 50 \sqrt{5} ({(- 2)}^{2} i^{2}) + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.15

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

$25 \sqrt{5} - 250 i + 50 \sqrt{5} (4 i^{2}) + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.16

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

$25 \sqrt{5} - 250 i + 50 \sqrt{5} (4 \cdot - 1) + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.17

Multiply $4$ by $- 1$ .

$25 \sqrt{5} - 250 i + 50 \sqrt{5} \cdot - 4 + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.18

Multiply $- 4$ by $50$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 10 {\sqrt{5}}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.19

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.1.19.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 10 {(5^{\frac{1}{2}})}^{2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.19.2

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 10 \cdot 5^{\frac{1}{2} \cdot 2} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.19.3

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 10 \cdot 5^{\frac{2}{2}} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.19.4

Cancel the common factor of $2$ .

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Step 2.1.19.4.1

Cancel the common factor.

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 10 \cdot 5^{\frac{2}{2}} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.19.4.2

Rewrite the expression.

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 10 \cdot 5^{1} {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.19.5

Evaluate the exponent.

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 10 \cdot 5 {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.20

Multiply $10$ by $5$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 50 {(- 2 i)}^{3} + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.21

Apply the product rule to $- 2 i$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 50 ({(- 2)}^{3} i^{3}) + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.22

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 50 (- 8 i^{3}) + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.23

Factor out $i^{2}$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 50 (- 8 (i^{2} \cdot i)) + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.24

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 50 (- 8 (- 1 \cdot i)) + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.25

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 50 (- 8 (- i)) + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.26

Multiply $- 1$ by $- 8$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 50 (8 i) + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.27

Multiply $8$ by $50$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 5 \sqrt{5} {(- 2 i)}^{4} + {(- 2 i)}^{5}$

Step 2.1.28

Apply the product rule to $- 2 i$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 5 \sqrt{5} ({(- 2)}^{4} i^{4}) + {(- 2 i)}^{5}$

Step 2.1.29

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 5 \sqrt{5} (16 i^{4}) + {(- 2 i)}^{5}$

Step 2.1.30

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

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Step 2.1.30.1

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 5 \sqrt{5} (16 {(i^{2})}^{2}) + {(- 2 i)}^{5}$

Step 2.1.30.2

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 5 \sqrt{5} (16 {(- 1)}^{2}) + {(- 2 i)}^{5}$

Step 2.1.30.3

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 5 \sqrt{5} (16 \cdot 1) + {(- 2 i)}^{5}$

Step 2.1.31

Multiply $16$ by $1$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 5 \sqrt{5} \cdot 16 + {(- 2 i)}^{5}$

Step 2.1.32

Multiply $16$ by $5$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 80 \sqrt{5} + {(- 2 i)}^{5}$

Step 2.1.33

Apply the product rule to $- 2 i$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 80 \sqrt{5} + {(- 2)}^{5} i^{5}$

Step 2.1.34

Raise $- 2$ to the power of $5$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 80 \sqrt{5} - 32 i^{5}$

Step 2.1.35

Factor out $i^{4}$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 80 \sqrt{5} - 32 (i^{4} i)$

Step 2.1.36

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

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Step 2.1.36.1

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 80 \sqrt{5} - 32 ({(i^{2})}^{2} i)$

Step 2.1.36.2

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 80 \sqrt{5} - 32 ({(- 1)}^{2} i)$

Step 2.1.36.3

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

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 80 \sqrt{5} - 32 (1 i)$

Step 2.1.37

Multiply $i$ by $1$ .

$25 \sqrt{5} - 250 i - 200 \sqrt{5} + 400 i + 80 \sqrt{5} - 32 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $200 \sqrt{5}$ from $25 \sqrt{5}$ .

$- 250 i - 175 \sqrt{5} + 400 i + 80 \sqrt{5} - 32 i$

Step 2.2.2

Add $- 250 i$ and $400 i$ .

$150 i - 175 \sqrt{5} + 80 \sqrt{5} - 32 i$

Step 2.2.3

Subtract $32 i$ from $150 i$ .

$118 i - 175 \sqrt{5} + 80 \sqrt{5}$

Step 2.2.4

Add $- 175 \sqrt{5}$ and $80 \sqrt{5}$ .

$118 i - 95 \sqrt{5}$

Step 2.2.5

Reorder $118 i$ and $- 95 \sqrt{5}$ .

$- 95 \sqrt{5} + 118 i$