Basic Math Examples

${(9 - 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1

Simplify each term.

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Step 1.1

Use the Binomial Theorem.

$9^{3} + 3 \cdot 9^{2} (- 4 \sqrt{5}) + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2

Simplify each term.

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Step 1.2.1

Raise $9$ to the power of $3$ .

$729 + 3 \cdot 9^{2} (- 4 \sqrt{5}) + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.2

Raise $9$ to the power of $2$ .

$729 + 3 \cdot 81 (- 4 \sqrt{5}) + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.3

Multiply $3$ by $81$ .

$729 + 243 (- 4 \sqrt{5}) + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.4

Multiply $- 4$ by $243$ .

$729 - 972 \sqrt{5} + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.5

Multiply $3$ by $9$ .

$729 - 972 \sqrt{5} + 27 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.6

Apply the product rule to $- 4 \sqrt{5}$ .

$729 - 972 \sqrt{5} + 27 ({(- 4)}^{2} {\sqrt{5}}^{2}) + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.7

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

$729 - 972 \sqrt{5} + 27 (16 {\sqrt{5}}^{2}) + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.8

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

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Step 1.2.8.1

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

$729 - 972 \sqrt{5} + 27 (16 {(5^{\frac{1}{2}})}^{2}) + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.8.2

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

$729 - 972 \sqrt{5} + 27 (16 \cdot 5^{\frac{1}{2} \cdot 2}) + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.8.3

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

$729 - 972 \sqrt{5} + 27 (16 \cdot 5^{\frac{2}{2}}) + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

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.

$729 - 972 \sqrt{5} + 27 (16 \cdot 5^{\frac{2}{2}}) + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.8.4.2

Rewrite the expression.

$729 - 972 \sqrt{5} + 27 (16 \cdot 5^{1}) + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.8.5

Evaluate the exponent.

$729 - 972 \sqrt{5} + 27 (16 \cdot 5) + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.9

Multiply $27 (16 \cdot 5)$ .

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Step 1.2.9.1

Multiply $16$ by $5$ .

$729 - 972 \sqrt{5} + 27 \cdot 80 + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.9.2

Multiply $27$ by $80$ .

$729 - 972 \sqrt{5} + 2160 + {(- 4 \sqrt{5})}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.10

Apply the product rule to $- 4 \sqrt{5}$ .

$729 - 972 \sqrt{5} + 2160 + {(- 4)}^{3} {\sqrt{5}}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.11

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

$729 - 972 \sqrt{5} + 2160 - 64 {\sqrt{5}}^{3} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.12

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

$729 - 972 \sqrt{5} + 2160 - 64 \sqrt{5^{3}} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.13

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

$729 - 972 \sqrt{5} + 2160 - 64 \sqrt{125} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.14

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

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Step 1.2.14.1

Factor $25$ out of $125$ .

$729 - 972 \sqrt{5} + 2160 - 64 \sqrt{25 (5)} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.14.2

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

$729 - 972 \sqrt{5} + 2160 - 64 \sqrt{5^{2} \cdot 5} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.15

Pull terms out from under the radical.

$729 - 972 \sqrt{5} + 2160 - 64 (5 \sqrt{5}) + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.2.16

Multiply $5$ by $- 64$ .

$729 - 972 \sqrt{5} + 2160 - 320 \sqrt{5} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.3

Add $729$ and $2160$ .

$2889 - 972 \sqrt{5} - 320 \sqrt{5} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.4

Subtract $320 \sqrt{5}$ from $- 972 \sqrt{5}$ .

$2889 - 1292 \sqrt{5} + 1 \div {(9 - 4 \sqrt{5})}^{3}$

Step 1.5

Rewrite the division as a fraction.

$2889 - 1292 \sqrt{5} + \frac{1}{{(9 - 4 \sqrt{5})}^{3}}$

Step 1.6

Use the Binomial Theorem.

$2889 - 1292 \sqrt{5} + \frac{1}{9^{3} + 3 \cdot 9^{2} (- 4 \sqrt{5}) + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3}}$

Step 1.7

Simplify each term.

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Step 1.7.1

Raise $9$ to the power of $3$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 + 3 \cdot 9^{2} (- 4 \sqrt{5}) + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.2

Raise $9$ to the power of $2$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 + 3 \cdot 81 (- 4 \sqrt{5}) + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.3

Multiply $3$ by $81$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 + 243 (- 4 \sqrt{5}) + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.4

Multiply $- 4$ by $243$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 3 \cdot 9 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.5

Multiply $3$ by $9$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 {(- 4 \sqrt{5})}^{2} + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.6

Apply the product rule to $- 4 \sqrt{5}$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 ({(- 4)}^{2} {\sqrt{5}}^{2}) + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.7

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

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 (16 {\sqrt{5}}^{2}) + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.8

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

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Step 1.7.8.1

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

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 (16 {(5^{\frac{1}{2}})}^{2}) + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.8.2

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

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 (16 \cdot 5^{\frac{1}{2} \cdot 2}) + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.8.3

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

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 (16 \cdot 5^{\frac{2}{2}}) + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.8.4

Cancel the common factor of $2$ .

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Step 1.7.8.4.1

Cancel the common factor.

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 (16 \cdot 5^{\frac{2}{2}}) + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.8.4.2

Rewrite the expression.

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 (16 \cdot 5^{1}) + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.8.5

Evaluate the exponent.

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 (16 \cdot 5) + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.9

Multiply $27 (16 \cdot 5)$ .

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Step 1.7.9.1

Multiply $16$ by $5$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 27 \cdot 80 + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.9.2

Multiply $27$ by $80$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 + {(- 4 \sqrt{5})}^{3}}$

Step 1.7.10

Apply the product rule to $- 4 \sqrt{5}$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 + {(- 4)}^{3} {\sqrt{5}}^{3}}$

Step 1.7.11

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

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 - 64 {\sqrt{5}}^{3}}$

Step 1.7.12

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

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 - 64 \sqrt{5^{3}}}$

Step 1.7.13

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

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 - 64 \sqrt{125}}$

Step 1.7.14

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

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Step 1.7.14.1

Factor $25$ out of $125$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 - 64 \sqrt{25 (5)}}$

Step 1.7.14.2

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

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 - 64 \sqrt{5^{2} \cdot 5}}$

Step 1.7.15

Pull terms out from under the radical.

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 - 64 (5 \sqrt{5})}$

Step 1.7.16

Multiply $5$ by $- 64$ .

$2889 - 1292 \sqrt{5} + \frac{1}{729 - 972 \sqrt{5} + 2160 - 320 \sqrt{5}}$

Step 1.8

Add $729$ and $2160$ .

$2889 - 1292 \sqrt{5} + \frac{1}{2889 - 972 \sqrt{5} - 320 \sqrt{5}}$

Step 1.9

Subtract $320 \sqrt{5}$ from $- 972 \sqrt{5}$ .

$2889 - 1292 \sqrt{5} + \frac{1}{2889 - 1292 \sqrt{5}}$

Step 1.10

Multiply $\frac{1}{2889 - 1292 \sqrt{5}}$ by $\frac{2889 + 1292 \sqrt{5}}{2889 + 1292 \sqrt{5}}$ .

$2889 - 1292 \sqrt{5} + \frac{1}{2889 - 1292 \sqrt{5}} \cdot \frac{2889 + 1292 \sqrt{5}}{2889 + 1292 \sqrt{5}}$

Step 1.11

Multiply $\frac{1}{2889 - 1292 \sqrt{5}}$ by $\frac{2889 + 1292 \sqrt{5}}{2889 + 1292 \sqrt{5}}$ .

$2889 - 1292 \sqrt{5} + \frac{2889 + 1292 \sqrt{5}}{(2889 - 1292 \sqrt{5}) (2889 + 1292 \sqrt{5})}$

Step 1.12

Expand the denominator using the FOIL method.

$2889 - 1292 \sqrt{5} + \frac{2889 + 1292 \sqrt{5}}{8346321 + 3732588 \sqrt{5} - 3732588 \sqrt{5} - 1669264 {\sqrt{5}}^{2}}$

Step 1.13

Simplify.

$2889 - 1292 \sqrt{5} + \frac{2889 + 1292 \sqrt{5}}{1}$

Step 1.14

Divide $2889 + 1292 \sqrt{5}$ by $1$ .

$2889 - 1292 \sqrt{5} + 2889 + 1292 \sqrt{5}$

Step 2

Simplify by adding terms.

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Step 2.1

Add $2889$ and $2889$ .

$5778 - 1292 \sqrt{5} + 1292 \sqrt{5}$

Step 2.2

Add $- 1292 \sqrt{5}$ and $1292 \sqrt{5}$ .

$5778 + 0$

Step 2.3

Add $5778$ and $0$ .

$5778$