Basic Math Examples

${(1 - \sqrt{3})}^{4}$

Step 1

Use the Binomial Theorem.

$1^{4} + 4 \cdot 1^{3} (- \sqrt{3}) + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

$1 + 4 \cdot 1^{3} (- \sqrt{3}) + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.2

One to any power is one.

$1 + 4 \cdot 1 (- \sqrt{3}) + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.3

Multiply $4$ by $1$ .

$1 + 4 (- \sqrt{3}) + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.4

Multiply $- 1$ by $4$ .

$1 - 4 \sqrt{3} + 6 \cdot 1^{2} {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.5

One to any power is one.

$1 - 4 \sqrt{3} + 6 \cdot 1 {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.6

Multiply $6$ by $1$ .

$1 - 4 \sqrt{3} + 6 {(- \sqrt{3})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.7

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

$1 - 4 \sqrt{3} + 6 ({(- 1)}^{2} {\sqrt{3}}^{2}) + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.8

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

$1 - 4 \sqrt{3} + 6 (1 {\sqrt{3}}^{2}) + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.9

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$1 - 4 \sqrt{3} + 6 {\sqrt{3}}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.10

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

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Step 2.1.10.1

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

$1 - 4 \sqrt{3} + 6 {(3^{\frac{1}{2}})}^{2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.10.2

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

$1 - 4 \sqrt{3} + 6 \cdot 3^{\frac{1}{2} \cdot 2} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.10.3

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

$1 - 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.10.4

Cancel the common factor of $2$ .

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Step 2.1.10.4.1

Cancel the common factor.

$1 - 4 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.10.4.2

Rewrite the expression.

$1 - 4 \sqrt{3} + 6 \cdot 3^{1} + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.10.5

Evaluate the exponent.

$1 - 4 \sqrt{3} + 6 \cdot 3 + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.11

Multiply $6$ by $3$ .

$1 - 4 \sqrt{3} + 18 + 4 \cdot 1 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.12

Multiply $4$ by $1$ .

$1 - 4 \sqrt{3} + 18 + 4 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4}$

Step 2.1.13

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

$1 - 4 \sqrt{3} + 18 + 4 ({(- 1)}^{3} {\sqrt{3}}^{3}) + {(- \sqrt{3})}^{4}$

Step 2.1.14

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

$1 - 4 \sqrt{3} + 18 + 4 (- {\sqrt{3}}^{3}) + {(- \sqrt{3})}^{4}$

Step 2.1.15

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

$1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{3^{3}}) + {(- \sqrt{3})}^{4}$

Step 2.1.16

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

$1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{27}) + {(- \sqrt{3})}^{4}$

Step 2.1.17

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.1.17.1

Factor $9$ out of $27$ .

$1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{9 (3)}) + {(- \sqrt{3})}^{4}$

Step 2.1.17.2

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

$1 - 4 \sqrt{3} + 18 + 4 (- \sqrt{3^{2} \cdot 3}) + {(- \sqrt{3})}^{4}$

Step 2.1.18

Pull terms out from under the radical.

$1 - 4 \sqrt{3} + 18 + 4 (- (3 \sqrt{3})) + {(- \sqrt{3})}^{4}$

Step 2.1.19

Multiply $3$ by $- 1$ .

$1 - 4 \sqrt{3} + 18 + 4 (- 3 \sqrt{3}) + {(- \sqrt{3})}^{4}$

Step 2.1.20

Multiply $- 3$ by $4$ .

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(- \sqrt{3})}^{4}$

Step 2.1.21

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

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(- 1)}^{4} {\sqrt{3}}^{4}$

Step 2.1.22

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

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 1 {\sqrt{3}}^{4}$

Step 2.1.23

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {\sqrt{3}}^{4}$

Step 2.1.24

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

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Step 2.1.24.1

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

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + {(3^{\frac{1}{2}})}^{4}$

Step 2.1.24.2

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

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{1}{2} \cdot 4}$

Step 2.1.24.3

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

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{4}{2}}$

Step 2.1.24.4

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

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Step 2.1.24.4.1

Factor $2$ out of $4$ .

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2}}$

Step 2.1.24.4.2

Cancel the common factors.

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Step 2.1.24.4.2.1

Factor $2$ out of $2$ .

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}}$

Step 2.1.24.4.2.2

Cancel the common factor.

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 2.1.24.4.2.3

Rewrite the expression.

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{\frac{2}{1}}$

Step 2.1.24.4.2.4

Divide $2$ by $1$ .

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 3^{2}$

Step 2.1.25

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

$1 - 4 \sqrt{3} + 18 - 12 \sqrt{3} + 9$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $1$ and $18$ .

$19 - 4 \sqrt{3} - 12 \sqrt{3} + 9$

Step 2.2.2

Add $19$ and $9$ .

$28 - 4 \sqrt{3} - 12 \sqrt{3}$

Step 2.2.3

Subtract $12 \sqrt{3}$ from $- 4 \sqrt{3}$ .

$28 - 16 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$28 - 16 \sqrt{3}$

Decimal Form:

$0.28718707 \dots$