Linear Algebra Examples

$(9 \sqrt{5} - 5 \sqrt{6}) (3 \sqrt{5} + 2 \sqrt{6})$

Step 1

Expand $(9 \sqrt{5} - 5 \sqrt{6}) (3 \sqrt{5} + 2 \sqrt{6})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$9 \sqrt{5} (3 \sqrt{5} + 2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5} + 2 \sqrt{6})$

Step 1.2

Apply the distributive property.

$9 \sqrt{5} (3 \sqrt{5}) + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5} + 2 \sqrt{6})$

Step 1.3

Apply the distributive property.

$9 \sqrt{5} (3 \sqrt{5}) + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $9 \sqrt{5} (3 \sqrt{5})$ .

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Step 2.1.1.1

Multiply $3$ by $9$ .

$27 \sqrt{5} \sqrt{5} + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.1.2

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

$27 ({\sqrt{5}}^{1} \sqrt{5}) + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.1.3

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

$27 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.1.4

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

$27 {\sqrt{5}}^{1 + 1} + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.1.5

Add $1$ and $1$ .

$27 {\sqrt{5}}^{2} + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.2

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

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Step 2.1.2.1

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

$27 {(5^{\frac{1}{2}})}^{2} + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.2.2

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

$27 \cdot 5^{\frac{1}{2} \cdot 2} + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.2.3

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

$27 \cdot 5^{\frac{2}{2}} + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

$27 \cdot 5^{\frac{2}{2}} + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.2.4.2

Rewrite the expression.

$27 \cdot 5^{1} + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.2.5

Evaluate the exponent.

$27 \cdot 5 + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.3

Multiply $27$ by $5$ .

$135 + 9 \sqrt{5} (2 \sqrt{6}) - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.4

Multiply $9 \sqrt{5} (2 \sqrt{6})$ .

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Step 2.1.4.1

Multiply $2$ by $9$ .

$135 + 18 \sqrt{5} \sqrt{6} - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.4.2

Combine using the product rule for radicals.

$135 + 18 \sqrt{6 \cdot 5} - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.4.3

Multiply $6$ by $5$ .

$135 + 18 \sqrt{30} - 5 \sqrt{6} (3 \sqrt{5}) - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.5

Multiply $- 5 \sqrt{6} (3 \sqrt{5})$ .

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Step 2.1.5.1

Multiply $3$ by $- 5$ .

$135 + 18 \sqrt{30} - 15 \sqrt{6} \sqrt{5} - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.5.2

Combine using the product rule for radicals.

$135 + 18 \sqrt{30} - 15 \sqrt{5 \cdot 6} - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.5.3

Multiply $5$ by $6$ .

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 5 \sqrt{6} (2 \sqrt{6})$

Step 2.1.6

Multiply $- 5 \sqrt{6} (2 \sqrt{6})$ .

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Step 2.1.6.1

Multiply $2$ by $- 5$ .

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 \sqrt{6} \sqrt{6}$

Step 2.1.6.2

Raise $\sqrt{6}$ to the power of $1$ .

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 ({\sqrt{6}}^{1} \sqrt{6})$

Step 2.1.6.3

Raise $\sqrt{6}$ to the power of $1$ .

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})$

Step 2.1.6.4

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

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 {\sqrt{6}}^{1 + 1}$

Step 2.1.6.5

Add $1$ and $1$ .

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 {\sqrt{6}}^{2}$

Step 2.1.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 {(6^{\frac{1}{2}})}^{2}$

Step 2.1.7.2

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

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 \cdot 6^{\frac{1}{2} \cdot 2}$

Step 2.1.7.3

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

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 \cdot 6^{\frac{2}{2}}$

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 \cdot 6^{\frac{2}{2}}$

Step 2.1.7.4.2

Rewrite the expression.

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 \cdot 6^{1}$

Step 2.1.7.5

Evaluate the exponent.

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 10 \cdot 6$

Step 2.1.8

Multiply $- 10$ by $6$ .

$135 + 18 \sqrt{30} - 15 \sqrt{30} - 60$

Step 2.2

Subtract $60$ from $135$ .

$75 + 18 \sqrt{30} - 15 \sqrt{30}$

Step 2.3

Subtract $15 \sqrt{30}$ from $18 \sqrt{30}$ .

$75 + 3 \sqrt{30}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$75 + 3 \sqrt{30}$

Decimal Form:

$91.43167672 \dots$