Algebra Examples

$\frac{- 7 \sqrt{2} + 4 \sqrt{3}}{3 \sqrt{2} - 5 \sqrt{3}}$

Step 1

Multiply $\frac{- 7 \sqrt{2} + 4 \sqrt{3}}{3 \sqrt{2} - 5 \sqrt{3}}$ by $\frac{3 \sqrt{2} + 5 \sqrt{3}}{3 \sqrt{2} + 5 \sqrt{3}}$ .

$\frac{- 7 \sqrt{2} + 4 \sqrt{3}}{3 \sqrt{2} - 5 \sqrt{3}} \cdot \frac{3 \sqrt{2} + 5 \sqrt{3}}{3 \sqrt{2} + 5 \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{- 7 \sqrt{2} + 4 \sqrt{3}}{3 \sqrt{2} - 5 \sqrt{3}}$ by $\frac{3 \sqrt{2} + 5 \sqrt{3}}{3 \sqrt{2} + 5 \sqrt{3}}$ .

$\frac{(- 7 \sqrt{2} + 4 \sqrt{3}) (3 \sqrt{2} + 5 \sqrt{3})}{(3 \sqrt{2} - 5 \sqrt{3}) (3 \sqrt{2} + 5 \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(- 7 \sqrt{2} + 4 \sqrt{3}) (3 \sqrt{2} + 5 \sqrt{3})}{9 {\sqrt{2}}^{2} + 15 \sqrt{6} - 15 \sqrt{6} - 25 {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(- 7 \sqrt{2} + 4 \sqrt{3}) (3 \sqrt{2} + 5 \sqrt{3})}{- 57}$

Step 3

Expand $(- 7 \sqrt{2} + 4 \sqrt{3}) (3 \sqrt{2} + 5 \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{- 7 \sqrt{2} (3 \sqrt{2} + 5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2} + 5 \sqrt{3})}{- 57}$

Step 3.2

Apply the distributive property.

$\frac{- 7 \sqrt{2} (3 \sqrt{2}) - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2} + 5 \sqrt{3})}{- 57}$

Step 3.3

Apply the distributive property.

$\frac{- 7 \sqrt{2} (3 \sqrt{2}) - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

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 $- 7 \sqrt{2} (3 \sqrt{2})$ .

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Step 4.1.1.1

Multiply $3$ by $- 7$ .

$\frac{- 21 \sqrt{2} \sqrt{2} - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.1.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{- 21 ({\sqrt{2}}^{1} \sqrt{2}) - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.1.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{- 21 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.1.4

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

$\frac{- 21 {\sqrt{2}}^{1 + 1} - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.1.5

Add $1$ and $1$ .

$\frac{- 21 {\sqrt{2}}^{2} - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.2

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

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Step 4.1.2.1

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

$\frac{- 21 {(2^{\frac{1}{2}})}^{2} - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.2.2

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

$\frac{- 21 \cdot 2^{\frac{1}{2} \cdot 2} - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.2.3

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

$\frac{- 21 \cdot 2^{\frac{2}{2}} - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

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.

$\frac{- 21 \cdot 2^{\frac{2}{2}} - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.2.4.2

Rewrite the expression.

$\frac{- 21 \cdot 2^{1} - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.2.5

Evaluate the exponent.

$\frac{- 21 \cdot 2 - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.3

Multiply $- 21$ by $2$ .

$\frac{- 42 - 7 \sqrt{2} (5 \sqrt{3}) + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.4

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

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Step 4.1.4.1

Multiply $5$ by $- 7$ .

$\frac{- 42 - 35 \sqrt{2} \sqrt{3} + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.4.2

Combine using the product rule for radicals.

$\frac{- 42 - 35 \sqrt{3 \cdot 2} + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.4.3

Multiply $3$ by $2$ .

$\frac{- 42 - 35 \sqrt{6} + 4 \sqrt{3} (3 \sqrt{2}) + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.5

Multiply $4 \sqrt{3} (3 \sqrt{2})$ .

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Step 4.1.5.1

Multiply $3$ by $4$ .

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{3} \sqrt{2} + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.5.2

Combine using the product rule for radicals.

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{2 \cdot 3} + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.5.3

Multiply $2$ by $3$ .

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 4 \sqrt{3} (5 \sqrt{3})}{- 57}$

Step 4.1.6

Multiply $4 \sqrt{3} (5 \sqrt{3})$ .

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Step 4.1.6.1

Multiply $5$ by $4$ .

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 \sqrt{3} \sqrt{3}}{- 57}$

Step 4.1.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 ({\sqrt{3}}^{1} \sqrt{3})}{- 57}$

Step 4.1.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{- 57}$

Step 4.1.6.4

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

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 {\sqrt{3}}^{1 + 1}}{- 57}$

Step 4.1.6.5

Add $1$ and $1$ .

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 {\sqrt{3}}^{2}}{- 57}$

Step 4.1.7

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

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Step 4.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 {(3^{\frac{1}{2}})}^{2}}{- 57}$

Step 4.1.7.2

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

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 \cdot 3^{\frac{1}{2} \cdot 2}}{- 57}$

Step 4.1.7.3

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

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 \cdot 3^{\frac{2}{2}}}{- 57}$

Step 4.1.7.4

Cancel the common factor of $2$ .

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Step 4.1.7.4.1

Cancel the common factor.

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 \cdot 3^{\frac{2}{2}}}{- 57}$

Step 4.1.7.4.2

Rewrite the expression.

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 \cdot 3^{1}}{- 57}$

Step 4.1.7.5

Evaluate the exponent.

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 20 \cdot 3}{- 57}$

Step 4.1.8

Multiply $20$ by $3$ .

$\frac{- 42 - 35 \sqrt{6} + 12 \sqrt{6} + 60}{- 57}$

Step 4.2

Add $- 42$ and $60$ .

$\frac{18 - 35 \sqrt{6} + 12 \sqrt{6}}{- 57}$

Step 4.3

Add $- 35 \sqrt{6}$ and $12 \sqrt{6}$ .

$\frac{18 - 23 \sqrt{6}}{- 57}$

Step 5

Move the negative in front of the fraction.

$- \frac{18 - 23 \sqrt{6}}{57}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{18 - 23 \sqrt{6}}{57}$

Decimal Form:

$0.67260112 \dots$