Basic Math Examples

${(3 \sqrt{7} - 5 \sqrt{2})}^{3}$

Step 1

Use the Binomial Theorem.

${(3 \sqrt{7})}^{3} + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $3 \sqrt{7}$ .

$3^{3} {\sqrt{7}}^{3} + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.2

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

$27 {\sqrt{7}}^{3} + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.3

Rewrite ${\sqrt{7}}^{3}$ as $\sqrt{7^{3}}$ .

$27 \sqrt{7^{3}} + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.4

Raise $7$ to the power of $3$ .

$27 \sqrt{343} + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.5

Rewrite $343$ as $7^{2} \cdot 7$ .

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Step 2.1.5.1

Factor $49$ out of $343$ .

$27 \sqrt{49 (7)} + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.5.2

Rewrite $49$ as $7^{2}$ .

$27 \sqrt{7^{2} \cdot 7} + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.6

Pull terms out from under the radical.

$27 (7 \sqrt{7}) + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.7

Multiply $7$ by $27$ .

$189 \sqrt{7} + 3 {(3 \sqrt{7})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.8

Apply the product rule to $3 \sqrt{7}$ .

$189 \sqrt{7} + 3 (3^{2} {\sqrt{7}}^{2}) (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.9

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

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Step 2.1.9.1

Move $3^{2}$ .

$189 \sqrt{7} + 3^{2} \cdot 3 {\sqrt{7}}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.9.2

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

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Step 2.1.9.2.1

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

$189 \sqrt{7} + 3^{2} \cdot 3^{1} {\sqrt{7}}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.9.2.2

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

$189 \sqrt{7} + 3^{2 + 1} {\sqrt{7}}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.9.3

Add $2$ and $1$ .

$189 \sqrt{7} + 3^{3} {\sqrt{7}}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.10

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

$189 \sqrt{7} + 27 {\sqrt{7}}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.11

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$189 \sqrt{7} + 27 {(7^{\frac{1}{2}})}^{2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.11.2

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

$189 \sqrt{7} + 27 \cdot 7^{\frac{1}{2} \cdot 2} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.11.3

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

$189 \sqrt{7} + 27 \cdot 7^{\frac{2}{2}} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.11.4

Cancel the common factor of $2$ .

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Step 2.1.11.4.1

Cancel the common factor.

$189 \sqrt{7} + 27 \cdot 7^{\frac{2}{2}} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.11.4.2

Rewrite the expression.

$189 \sqrt{7} + 27 \cdot 7^{1} (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.11.5

Evaluate the exponent.

$189 \sqrt{7} + 27 \cdot 7 (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.12

Multiply $27$ by $7$ .

$189 \sqrt{7} + 189 (- 5 \sqrt{2}) + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.13

Multiply $- 5$ by $189$ .

$189 \sqrt{7} - 945 \sqrt{2} + 3 (3 \sqrt{7}) {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.14

Multiply $3$ by $3$ .

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} {(- 5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.15

Apply the product rule to $- 5 \sqrt{2}$ .

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} ({(- 5)}^{2} {\sqrt{2}}^{2}) + {(- 5 \sqrt{2})}^{3}$

Step 2.1.16

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

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} (25 {\sqrt{2}}^{2}) + {(- 5 \sqrt{2})}^{3}$

Step 2.1.17

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

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Step 2.1.17.1

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

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} (25 {(2^{\frac{1}{2}})}^{2}) + {(- 5 \sqrt{2})}^{3}$

Step 2.1.17.2

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

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} (25 \cdot 2^{\frac{1}{2} \cdot 2}) + {(- 5 \sqrt{2})}^{3}$

Step 2.1.17.3

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

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} (25 \cdot 2^{\frac{2}{2}}) + {(- 5 \sqrt{2})}^{3}$

Step 2.1.17.4

Cancel the common factor of $2$ .

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Step 2.1.17.4.1

Cancel the common factor.

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} (25 \cdot 2^{\frac{2}{2}}) + {(- 5 \sqrt{2})}^{3}$

Step 2.1.17.4.2

Rewrite the expression.

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} (25 \cdot 2^{1}) + {(- 5 \sqrt{2})}^{3}$

Step 2.1.17.5

Evaluate the exponent.

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} (25 \cdot 2) + {(- 5 \sqrt{2})}^{3}$

Step 2.1.18

Multiply $25$ by $2$ .

$189 \sqrt{7} - 945 \sqrt{2} + 9 \sqrt{7} \cdot 50 + {(- 5 \sqrt{2})}^{3}$

Step 2.1.19

Multiply $50$ by $9$ .

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} + {(- 5 \sqrt{2})}^{3}$

Step 2.1.20

Apply the product rule to $- 5 \sqrt{2}$ .

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} + {(- 5)}^{3} {\sqrt{2}}^{3}$

Step 2.1.21

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

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} - 125 {\sqrt{2}}^{3}$

Step 2.1.22

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

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} - 125 \sqrt{2^{3}}$

Step 2.1.23

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

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} - 125 \sqrt{8}$

Step 2.1.24

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

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Step 2.1.24.1

Factor $4$ out of $8$ .

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} - 125 \sqrt{4 (2)}$

Step 2.1.24.2

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

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} - 125 \sqrt{2^{2} \cdot 2}$

Step 2.1.25

Pull terms out from under the radical.

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} - 125 (2 \sqrt{2})$

Step 2.1.26

Multiply $2$ by $- 125$ .

$189 \sqrt{7} - 945 \sqrt{2} + 450 \sqrt{7} - 250 \sqrt{2}$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $189 \sqrt{7}$ and $450 \sqrt{7}$ .

$639 \sqrt{7} - 945 \sqrt{2} - 250 \sqrt{2}$

Step 2.2.2

Subtract $250 \sqrt{2}$ from $- 945 \sqrt{2}$ .

$639 \sqrt{7} - 1195 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$639 \sqrt{7} - 1195 \sqrt{2}$

Decimal Form:

$0.64988073 \dots$