Basic Math Examples

$\frac{\sqrt{5} + \sqrt{7}}{3 \sqrt{5} - \sqrt{7}}$

Step 1

Multiply $\frac{\sqrt{5} + \sqrt{7}}{3 \sqrt{5} - \sqrt{7}}$ by $\frac{3 \sqrt{5} + \sqrt{7}}{3 \sqrt{5} + \sqrt{7}}$ .

$\frac{\sqrt{5} + \sqrt{7}}{3 \sqrt{5} - \sqrt{7}} \cdot \frac{3 \sqrt{5} + \sqrt{7}}{3 \sqrt{5} + \sqrt{7}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{5} + \sqrt{7}}{3 \sqrt{5} - \sqrt{7}}$ by $\frac{3 \sqrt{5} + \sqrt{7}}{3 \sqrt{5} + \sqrt{7}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} + \sqrt{7}) (3 \sqrt{5} + \sqrt{7})}{9 {\sqrt{5}}^{2} + 3 \sqrt{35} - 3 \sqrt{35} - {\sqrt{7}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{5} + \sqrt{7}) (3 \sqrt{5} + \sqrt{7})}{38}$

Step 3

Expand $(\sqrt{5} + \sqrt{7}) (3 \sqrt{5} + \sqrt{7})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{5} (3 \sqrt{5} + \sqrt{7}) + \sqrt{7} (3 \sqrt{5} + \sqrt{7})}{38}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{5} (3 \sqrt{5}) + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5} + \sqrt{7})}{38}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{5} (3 \sqrt{5}) + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $\sqrt{5} (3 \sqrt{5})$ .

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Step 4.1.1.1

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 ({\sqrt{5}}^{1} \sqrt{5}) + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.1.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.1.3

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

$\frac{3 {\sqrt{5}}^{1 + 1} + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.1.4

Add $1$ and $1$ .

$\frac{3 {\sqrt{5}}^{2} + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.2

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

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Step 4.1.2.1

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

$\frac{3 {(5^{\frac{1}{2}})}^{2} + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.2.2

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

$\frac{3 \cdot 5^{\frac{1}{2} \cdot 2} + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.2.3

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

$\frac{3 \cdot 5^{\frac{2}{2}} + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

$\frac{3 \cdot 5^{\frac{2}{2}} + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.2.4.2

Rewrite the expression.

$\frac{3 \cdot 5^{1} + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.2.5

Evaluate the exponent.

$\frac{3 \cdot 5 + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.3

Multiply $3$ by $5$ .

$\frac{15 + \sqrt{5} \sqrt{7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.4

Combine using the product rule for radicals.

$\frac{15 + \sqrt{5 \cdot 7} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.5

Multiply $5$ by $7$ .

$\frac{15 + \sqrt{35} + \sqrt{7} (3 \sqrt{5}) + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.6

Multiply $\sqrt{7} (3 \sqrt{5})$ .

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Step 4.1.6.1

Combine using the product rule for radicals.

$\frac{15 + \sqrt{35} + 3 \sqrt{7 \cdot 5} + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.6.2

Multiply $7$ by $5$ .

$\frac{15 + \sqrt{35} + 3 \sqrt{35} + \sqrt{7} \sqrt{7}}{38}$

Step 4.1.7

Combine using the product rule for radicals.

$\frac{15 + \sqrt{35} + 3 \sqrt{35} + \sqrt{7 \cdot 7}}{38}$

Step 4.1.8

Multiply $7$ by $7$ .

$\frac{15 + \sqrt{35} + 3 \sqrt{35} + \sqrt{49}}{38}$

Step 4.1.9

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

$\frac{15 + \sqrt{35} + 3 \sqrt{35} + \sqrt{7^{2}}}{38}$

Step 4.1.10

Pull terms out from under the radical, assuming positive real numbers.

$\frac{15 + \sqrt{35} + 3 \sqrt{35} + 7}{38}$

Step 4.2

Add $15$ and $7$ .

$\frac{22 + \sqrt{35} + 3 \sqrt{35}}{38}$

Step 4.3

Add $\sqrt{35}$ and $3 \sqrt{35}$ .

$\frac{22 + 4 \sqrt{35}}{38}$

Step 5

Cancel the common factor of $22 + 4 \sqrt{35}$ and $38$ .

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Step 5.1

Factor $2$ out of $22$ .

$\frac{2 (11) + 4 \sqrt{35}}{38}$

Step 5.2

Factor $2$ out of $4 \sqrt{35}$ .

$\frac{2 (11) + 2 (2 \sqrt{35})}{38}$

Step 5.3

Factor $2$ out of $2 (11) + 2 (2 \sqrt{35})$ .

$\frac{2 (11 + 2 \sqrt{35})}{38}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $2$ out of $38$ .

$\frac{2 (11 + 2 \sqrt{35})}{2 \cdot 19}$

Step 5.4.2

Cancel the common factor.

$\frac{2 (11 + 2 \sqrt{35})}{2 \cdot 19}$

Step 5.4.3

Rewrite the expression.

$\frac{11 + 2 \sqrt{35}}{19}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{11 + 2 \sqrt{35}}{19}$

Decimal Form:

$1.20169260 \dots$