Trigonometry Examples

$\frac{5 \sqrt{28} - 10 \sqrt{35}}{5 \sqrt{7}}$

Step 1

Cancel the common factor of $5 \sqrt{28} - 10 \sqrt{35}$ and $5$ .

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Step 1.1

Factor $5$ out of $5 \sqrt{28}$ .

$\frac{5 (\sqrt{28}) - 10 \sqrt{35}}{5 \sqrt{7}}$

Step 1.2

Factor $5$ out of $- 10 \sqrt{35}$ .

$\frac{5 (\sqrt{28}) + 5 (- 2 \sqrt{35})}{5 \sqrt{7}}$

Step 1.3

Factor $5$ out of $5 (\sqrt{28}) + 5 (- 2 \sqrt{35})$ .

$\frac{5 (\sqrt{28} - 2 \sqrt{35})}{5 \sqrt{7}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $5$ out of $5 \sqrt{7}$ .

$\frac{5 (\sqrt{28} - 2 \sqrt{35})}{5 (\sqrt{7})}$

Step 1.4.2

Cancel the common factor.

$\frac{5 (\sqrt{28} - 2 \sqrt{35})}{5 \sqrt{7}}$

Step 1.4.3

Rewrite the expression.

$\frac{\sqrt{28} - 2 \sqrt{35}}{\sqrt{7}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 2.1.1

Factor $4$ out of $28$ .

$\frac{\sqrt{4 (7)} - 2 \sqrt{35}}{\sqrt{7}}$

Step 2.1.2

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

$\frac{\sqrt{2^{2} \cdot 7} - 2 \sqrt{35}}{\sqrt{7}}$

Step 2.2

Pull terms out from under the radical.

$\frac{2 \sqrt{7} - 2 \sqrt{35}}{\sqrt{7}}$

Step 3

Multiply $\frac{2 \sqrt{7} - 2 \sqrt{35}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{2 \sqrt{7} - 2 \sqrt{35}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{2 \sqrt{7} - 2 \sqrt{35}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 4.4

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

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 4.6

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

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Step 4.6.1

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

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{7^{1}}$

Step 4.6.5

Evaluate the exponent.

$\frac{(2 \sqrt{7} - 2 \sqrt{35}) \sqrt{7}}{7}$

Step 5

Apply the distributive property.

$\frac{2 \sqrt{7} \sqrt{7} - 2 \sqrt{35} \sqrt{7}}{7}$

Step 6

Multiply $2 \sqrt{7} \sqrt{7}$ .

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Step 6.1

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{2 ({\sqrt{7}}^{1} \sqrt{7}) - 2 \sqrt{35} \sqrt{7}}{7}$

Step 6.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{2 ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) - 2 \sqrt{35} \sqrt{7}}{7}$

Step 6.3

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

$\frac{2 {\sqrt{7}}^{1 + 1} - 2 \sqrt{35} \sqrt{7}}{7}$

Step 6.4

Add $1$ and $1$ .

$\frac{2 {\sqrt{7}}^{2} - 2 \sqrt{35} \sqrt{7}}{7}$

Step 7

Multiply $- 2 \sqrt{35} \sqrt{7}$ .

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Step 7.1

Combine using the product rule for radicals.

$\frac{2 {\sqrt{7}}^{2} - 2 \sqrt{7 \cdot 35}}{7}$

Step 7.2

Multiply $7$ by $35$ .

$\frac{2 {\sqrt{7}}^{2} - 2 \sqrt{245}}{7}$

Step 8

Simplify each term.

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Step 8.1

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

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Step 8.1.1

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

$\frac{2 {(7^{\frac{1}{2}})}^{2} - 2 \sqrt{245}}{7}$

Step 8.1.2

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

$\frac{2 \cdot 7^{\frac{1}{2} \cdot 2} - 2 \sqrt{245}}{7}$

Step 8.1.3

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

$\frac{2 \cdot 7^{\frac{2}{2}} - 2 \sqrt{245}}{7}$

Step 8.1.4

Cancel the common factor of $2$ .

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Step 8.1.4.1

Cancel the common factor.

$\frac{2 \cdot 7^{\frac{2}{2}} - 2 \sqrt{245}}{7}$

Step 8.1.4.2

Rewrite the expression.

$\frac{2 \cdot 7^{1} - 2 \sqrt{245}}{7}$

Step 8.1.5

Evaluate the exponent.

$\frac{2 \cdot 7 - 2 \sqrt{245}}{7}$

Step 8.2

Multiply $2$ by $7$ .

$\frac{14 - 2 \sqrt{245}}{7}$

Step 8.3

Rewrite $245$ as $7^{2} \cdot 5$ .

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Step 8.3.1

Factor $49$ out of $245$ .

$\frac{14 - 2 \sqrt{49 (5)}}{7}$

Step 8.3.2

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

$\frac{14 - 2 \sqrt{7^{2} \cdot 5}}{7}$

Step 8.4

Pull terms out from under the radical.

$\frac{14 - 2 (7 \sqrt{5})}{7}$

Step 8.5

Multiply $7$ by $- 2$ .

$\frac{14 - 14 \sqrt{5}}{7}$

Step 9

Cancel the common factor of $14 - 14 \sqrt{5}$ and $7$ .

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Step 9.1

Factor $7$ out of $14$ .

$\frac{7 \cdot 2 - 14 \sqrt{5}}{7}$

Step 9.2

Factor $7$ out of $- 14 \sqrt{5}$ .

$\frac{7 \cdot 2 + 7 (- 2 \sqrt{5})}{7}$

Step 9.3

Factor $7$ out of $7 (2) + 7 (- 2 \sqrt{5})$ .

$\frac{7 (2 - 2 \sqrt{5})}{7}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $7$ out of $7$ .

$\frac{7 (2 - 2 \sqrt{5})}{7 (1)}$

Step 9.4.2

Cancel the common factor.

$\frac{7 (2 - 2 \sqrt{5})}{7 \cdot 1}$

Step 9.4.3

Rewrite the expression.

$\frac{2 - 2 \sqrt{5}}{1}$

Step 9.4.4

Divide $2 - 2 \sqrt{5}$ by $1$ .

$2 - 2 \sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$2 - 2 \sqrt{5}$

Decimal Form:

$- 2.47213595 \dots$