Basic Math Examples

${(\frac{2 - 2 i}{\sqrt{8}})}^{80}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

${(\frac{2 - 2 i}{\sqrt{4 (2)}})}^{80}$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

${(\frac{2 - 2 i}{\sqrt{2^{2} \cdot 2}})}^{80}$

Step 1.2

Pull terms out from under the radical.

${(\frac{2 - 2 i}{2 \sqrt{2}})}^{80}$

Step 2

Cancel the common factor of $2 - 2 i$ and $2$ .

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Step 2.1

Factor $2$ out of $2$ .

${(\frac{2 \cdot 1 - 2 i}{2 \sqrt{2}})}^{80}$

Step 2.2

Factor $2$ out of $- 2 i$ .

${(\frac{2 \cdot 1 + 2 (- i)}{2 \sqrt{2}})}^{80}$

Step 2.3

Factor $2$ out of $2 (1) + 2 (- i)$ .

${(\frac{2 (1 - i)}{2 \sqrt{2}})}^{80}$

Step 2.4

Cancel the common factors.

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Step 2.4.1

Factor $2$ out of $2 \sqrt{2}$ .

${(\frac{2 (1 - i)}{2 (\sqrt{2})})}^{80}$

Step 2.4.2

Cancel the common factor.

${(\frac{2 (1 - i)}{2 \sqrt{2}})}^{80}$

Step 2.4.3

Rewrite the expression.

${(\frac{1 - i}{\sqrt{2}})}^{80}$

Step 3

Multiply $\frac{1 - i}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{1 - i}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{80}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1 - i}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{(1 - i) \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{80}$

Step 4.2

Raise $\sqrt{2}$ to the power of $1$ .

${(\frac{(1 - i) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{80}$

Step 4.3

Raise $\sqrt{2}$ to the power of $1$ .

${(\frac{(1 - i) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{80}$

Step 4.4

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

${(\frac{(1 - i) \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{80}$

Step 4.5

Add $1$ and $1$ .

${(\frac{(1 - i) \sqrt{2}}{{\sqrt{2}}^{2}})}^{80}$

Step 4.6

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

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Step 4.6.1

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

${(\frac{(1 - i) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{80}$

Step 4.6.2

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

${(\frac{(1 - i) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{80}$

Step 4.6.3

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

${(\frac{(1 - i) \sqrt{2}}{2^{\frac{2}{2}}})}^{80}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

${(\frac{(1 - i) \sqrt{2}}{2^{\frac{2}{2}}})}^{80}$

Step 4.6.4.2

Rewrite the expression.

${(\frac{(1 - i) \sqrt{2}}{2^{1}})}^{80}$

Step 4.6.5

Evaluate the exponent.

${(\frac{(1 - i) \sqrt{2}}{2})}^{80}$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{(1 - i) \sqrt{2}}{2}$ .

$\frac{{((1 - i) \sqrt{2})}^{80}}{2^{80}}$

Step 5.2

Apply the product rule to $(1 - i) \sqrt{2}$ .

$\frac{{(1 - i)}^{80} {\sqrt{2}}^{80}}{2^{80}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt{2}}^{80}$ as $2^{40}$ .

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Step 6.1.1

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

$\frac{{(1 - i)}^{80} {(2^{\frac{1}{2}})}^{80}}{2^{80}}$

Step 6.1.2

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

$\frac{{(1 - i)}^{80} \cdot 2^{\frac{1}{2} \cdot 80}}{2^{80}}$

Step 6.1.3

Combine $\frac{1}{2}$ and $80$ .

$\frac{{(1 - i)}^{80} \cdot 2^{\frac{80}{2}}}{2^{80}}$

Step 6.1.4

Cancel the common factor of $80$ and $2$ .

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Step 6.1.4.1

Factor $2$ out of $80$ .

$\frac{{(1 - i)}^{80} \cdot 2^{\frac{2 \cdot 40}{2}}}{2^{80}}$

Step 6.1.4.2

Cancel the common factors.

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Step 6.1.4.2.1

Factor $2$ out of $2$ .

$\frac{{(1 - i)}^{80} \cdot 2^{\frac{2 \cdot 40}{2 (1)}}}{2^{80}}$

Step 6.1.4.2.2

Cancel the common factor.

$\frac{{(1 - i)}^{80} \cdot 2^{\frac{2 \cdot 40}{2 \cdot 1}}}{2^{80}}$

Step 6.1.4.2.3

Rewrite the expression.

$\frac{{(1 - i)}^{80} \cdot 2^{\frac{40}{1}}}{2^{80}}$

Step 6.1.4.2.4

Divide $40$ by $1$ .

$\frac{{(1 - i)}^{80} \cdot 2^{40}}{2^{80}}$

Step 6.2

Raise $2$ to the power of $40$ .

$\frac{{(1 - i)}^{80} \cdot 1099511627776}{2^{80}}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Raise $2$ to the power of $80$ .

$\frac{{(1 - i)}^{80} \cdot 1099511627776}{1208925819614630000000000}$

Step 7.2

Move $1099511627776$ to the left of ${(1 - i)}^{80}$ .

$\frac{1099511627776 {(1 - i)}^{80}}{1208925819614630000000000}$

Step 7.3

Cancel the common factor of $1099511627776$ and $1208925819614630000000000$ .

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Step 7.3.1

Factor $1024$ out of $1099511627776 {(1 - i)}^{80}$ .

$\frac{1024 (1073741824 {(1 - i)}^{80})}{1208925819614630000000000}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $1024$ out of $1208925819614630000000000$ .

$\frac{1024 (1073741824 {(1 - i)}^{80})}{1024 (1180591620717412109375)}$

Step 7.3.2.2

Cancel the common factor.

$\frac{1024 (1073741824 {(1 - i)}^{80})}{1024 \cdot 1180591620717412109375}$

Step 7.3.2.3

Rewrite the expression.

$\frac{1073741824 {(1 - i)}^{80}}{1180591620717412109375}$