Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

$\frac{1}{27 + 3 \cdot 3^{2} (- 2 \sqrt{2}) + 3 \cdot 3 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 1.2.2.1

Multiply $3$ by $3^{2}$ .

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Step 1.2.2.1.1

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

$\frac{1}{27 + 3^{1} \cdot 3^{2} (- 2 \sqrt{2}) + 3 \cdot 3 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.2.1.2

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

$\frac{1}{27 + 3^{1 + 2} (- 2 \sqrt{2}) + 3 \cdot 3 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.2.2

Add $1$ and $2$ .

$\frac{1}{27 + 3^{3} (- 2 \sqrt{2}) + 3 \cdot 3 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.3

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

$\frac{1}{27 + 27 (- 2 \sqrt{2}) + 3 \cdot 3 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.4

Multiply $- 2$ by $27$ .

$\frac{1}{27 - 54 \sqrt{2} + 3 \cdot 3 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.5

Multiply $3$ by $3$ .

$\frac{1}{27 - 54 \sqrt{2} + 9 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.6

Apply the product rule to $- 2 \sqrt{2}$ .

$\frac{1}{27 - 54 \sqrt{2} + 9 ({(- 2)}^{2} {\sqrt{2}}^{2}) + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.7

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

$\frac{1}{27 - 54 \sqrt{2} + 9 (4 {\sqrt{2}}^{2}) + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.8

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

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Step 1.2.8.1

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

$\frac{1}{27 - 54 \sqrt{2} + 9 (4 {(2^{\frac{1}{2}})}^{2}) + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.8.2

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

$\frac{1}{27 - 54 \sqrt{2} + 9 (4 \cdot 2^{\frac{1}{2} \cdot 2}) + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.8.3

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

$\frac{1}{27 - 54 \sqrt{2} + 9 (4 \cdot 2^{\frac{2}{2}}) + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.8.4

Cancel the common factor of $2$ .

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Step 1.2.8.4.1

Cancel the common factor.

$\frac{1}{27 - 54 \sqrt{2} + 9 (4 \cdot 2^{\frac{2}{2}}) + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.8.4.2

Rewrite the expression.

$\frac{1}{27 - 54 \sqrt{2} + 9 (4 \cdot 2^{1}) + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.8.5

Evaluate the exponent.

$\frac{1}{27 - 54 \sqrt{2} + 9 (4 \cdot 2) + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.9

Multiply $9 (4 \cdot 2)$ .

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Step 1.2.9.1

Multiply $4$ by $2$ .

$\frac{1}{27 - 54 \sqrt{2} + 9 \cdot 8 + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.9.2

Multiply $9$ by $8$ .

$\frac{1}{27 - 54 \sqrt{2} + 72 + {(- 2 \sqrt{2})}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.10

Apply the product rule to $- 2 \sqrt{2}$ .

$\frac{1}{27 - 54 \sqrt{2} + 72 + {(- 2)}^{3} {\sqrt{2}}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.11

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

$\frac{1}{27 - 54 \sqrt{2} + 72 - 8 {\sqrt{2}}^{3}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.12

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

$\frac{1}{27 - 54 \sqrt{2} + 72 - 8 \sqrt{2^{3}}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.13

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

$\frac{1}{27 - 54 \sqrt{2} + 72 - 8 \sqrt{8}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.14

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

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Step 1.2.14.1

Factor $4$ out of $8$ .

$\frac{1}{27 - 54 \sqrt{2} + 72 - 8 \sqrt{4 (2)}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.14.2

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

$\frac{1}{27 - 54 \sqrt{2} + 72 - 8 \sqrt{2^{2} \cdot 2}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.15

Pull terms out from under the radical.

$\frac{1}{27 - 54 \sqrt{2} + 72 - 8 (2 \sqrt{2})} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.2.16

Multiply $2$ by $- 8$ .

$\frac{1}{27 - 54 \sqrt{2} + 72 - 16 \sqrt{2}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.3

Add $27$ and $72$ .

$\frac{1}{99 - 54 \sqrt{2} - 16 \sqrt{2}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.4

Subtract $16 \sqrt{2}$ from $- 54 \sqrt{2}$ .

$\frac{1}{99 - 70 \sqrt{2}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.5

Multiply $\frac{1}{99 - 70 \sqrt{2}}$ by $\frac{99 + 70 \sqrt{2}}{99 + 70 \sqrt{2}}$ .

$\frac{1}{99 - 70 \sqrt{2}} \cdot \frac{99 + 70 \sqrt{2}}{99 + 70 \sqrt{2}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.6

Multiply $\frac{1}{99 - 70 \sqrt{2}}$ by $\frac{99 + 70 \sqrt{2}}{99 + 70 \sqrt{2}}$ .

$\frac{99 + 70 \sqrt{2}}{(99 - 70 \sqrt{2}) (99 + 70 \sqrt{2})} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.7

Expand the denominator using the FOIL method.

$\frac{99 + 70 \sqrt{2}}{9801 + 6930 \sqrt{2} - 6930 \sqrt{2} - 4900 {\sqrt{2}}^{2}} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.8

Simplify.

$\frac{99 + 70 \sqrt{2}}{1} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.9

Divide $99 + 70 \sqrt{2}$ by $1$ .

$99 + 70 \sqrt{2} + \frac{1}{{(3 + 2 \sqrt{2})}^{3}}$

Step 1.10

Use the Binomial Theorem.

$99 + 70 \sqrt{2} + \frac{1}{3^{3} + 3 \cdot 3^{2} (2 \sqrt{2}) + 3 \cdot 3 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3}}$

Step 1.11

Simplify each term.

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Step 1.11.1

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 3 \cdot 3^{2} (2 \sqrt{2}) + 3 \cdot 3 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3}}$

Step 1.11.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 1.11.2.1

Multiply $3$ by $3^{2}$ .

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Step 1.11.2.1.1

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 3^{1} \cdot 3^{2} (2 \sqrt{2}) + 3 \cdot 3 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3}}$

Step 1.11.2.1.2

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 3^{1 + 2} (2 \sqrt{2}) + 3 \cdot 3 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3}}$

Step 1.11.2.2

Add $1$ and $2$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 3^{3} (2 \sqrt{2}) + 3 \cdot 3 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3}}$

Step 1.11.3

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 27 (2 \sqrt{2}) + 3 \cdot 3 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3}}$

Step 1.11.4

Multiply $2$ by $27$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 3 \cdot 3 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3}}$

Step 1.11.5

Multiply $3$ by $3$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 {(2 \sqrt{2})}^{2} + {(2 \sqrt{2})}^{3}}$

Step 1.11.6

Apply the product rule to $2 \sqrt{2}$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 (2^{2} {\sqrt{2}}^{2}) + {(2 \sqrt{2})}^{3}}$

Step 1.11.7

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 (4 {\sqrt{2}}^{2}) + {(2 \sqrt{2})}^{3}}$

Step 1.11.8

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

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Step 1.11.8.1

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 (4 {(2^{\frac{1}{2}})}^{2}) + {(2 \sqrt{2})}^{3}}$

Step 1.11.8.2

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 (4 \cdot 2^{\frac{1}{2} \cdot 2}) + {(2 \sqrt{2})}^{3}}$

Step 1.11.8.3

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 (4 \cdot 2^{\frac{2}{2}}) + {(2 \sqrt{2})}^{3}}$

Step 1.11.8.4

Cancel the common factor of $2$ .

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Step 1.11.8.4.1

Cancel the common factor.

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 (4 \cdot 2^{\frac{2}{2}}) + {(2 \sqrt{2})}^{3}}$

Step 1.11.8.4.2

Rewrite the expression.

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 (4 \cdot 2^{1}) + {(2 \sqrt{2})}^{3}}$

Step 1.11.8.5

Evaluate the exponent.

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 (4 \cdot 2) + {(2 \sqrt{2})}^{3}}$

Step 1.11.9

Multiply $9 (4 \cdot 2)$ .

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Step 1.11.9.1

Multiply $4$ by $2$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 9 \cdot 8 + {(2 \sqrt{2})}^{3}}$

Step 1.11.9.2

Multiply $9$ by $8$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + {(2 \sqrt{2})}^{3}}$

Step 1.11.10

Apply the product rule to $2 \sqrt{2}$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + 2^{3} {\sqrt{2}}^{3}}$

Step 1.11.11

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + 8 {\sqrt{2}}^{3}}$

Step 1.11.12

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + 8 \sqrt{2^{3}}}$

Step 1.11.13

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + 8 \sqrt{8}}$

Step 1.11.14

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

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Step 1.11.14.1

Factor $4$ out of $8$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + 8 \sqrt{4 (2)}}$

Step 1.11.14.2

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

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + 8 \sqrt{2^{2} \cdot 2}}$

Step 1.11.15

Pull terms out from under the radical.

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + 8 (2 \sqrt{2})}$

Step 1.11.16

Multiply $2$ by $8$ .

$99 + 70 \sqrt{2} + \frac{1}{27 + 54 \sqrt{2} + 72 + 16 \sqrt{2}}$

Step 1.12

Add $27$ and $72$ .

$99 + 70 \sqrt{2} + \frac{1}{99 + 54 \sqrt{2} + 16 \sqrt{2}}$

Step 1.13

Add $54 \sqrt{2}$ and $16 \sqrt{2}$ .

$99 + 70 \sqrt{2} + \frac{1}{99 + 70 \sqrt{2}}$

Step 1.14

Multiply $\frac{1}{99 + 70 \sqrt{2}}$ by $\frac{99 - 70 \sqrt{2}}{99 - 70 \sqrt{2}}$ .

$99 + 70 \sqrt{2} + \frac{1}{99 + 70 \sqrt{2}} \cdot \frac{99 - 70 \sqrt{2}}{99 - 70 \sqrt{2}}$

Step 1.15

Multiply $\frac{1}{99 + 70 \sqrt{2}}$ by $\frac{99 - 70 \sqrt{2}}{99 - 70 \sqrt{2}}$ .

$99 + 70 \sqrt{2} + \frac{99 - 70 \sqrt{2}}{(99 + 70 \sqrt{2}) (99 - 70 \sqrt{2})}$

Step 1.16

Expand the denominator using the FOIL method.

$99 + 70 \sqrt{2} + \frac{99 - 70 \sqrt{2}}{9801 - 6930 \sqrt{2} + 6930 \sqrt{2} - 4900 {\sqrt{2}}^{2}}$

Step 1.17

Simplify.

$99 + 70 \sqrt{2} + \frac{99 - 70 \sqrt{2}}{1}$

Step 1.18

Divide $99 - 70 \sqrt{2}$ by $1$ .

$99 + 70 \sqrt{2} + 99 - 70 \sqrt{2}$

Step 2

Simplify by adding terms.

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Step 2.1

Add $99$ and $99$ .

$198 + 70 \sqrt{2} - 70 \sqrt{2}$

Step 2.2

Subtract $70 \sqrt{2}$ from $70 \sqrt{2}$ .

$198 + 0$

Step 2.3

Add $198$ and $0$ .

$198$