Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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 $2 \sqrt{5} (2 \sqrt{5})$ .

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Step 2.1.1.1

Multiply $2$ by $2$ .

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

Step 2.1.1.2

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

$4 ({\sqrt{5}}^{1} \sqrt{5}) + 2 \sqrt{5} (3 \sqrt{3}) - 3 \sqrt{3} (2 \sqrt{5}) - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.1.3

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

$4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 2 \sqrt{5} (3 \sqrt{3}) - 3 \sqrt{3} (2 \sqrt{5}) - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.1.4

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

$4 {\sqrt{5}}^{1 + 1} + 2 \sqrt{5} (3 \sqrt{3}) - 3 \sqrt{3} (2 \sqrt{5}) - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.1.5

Add $1$ and $1$ .

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

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 $5^{\frac{1}{2}}$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

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.

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

Step 2.1.2.4.2

Rewrite the expression.

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

Step 2.1.2.5

Evaluate the exponent.

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

Step 2.1.3

Multiply $4$ by $5$ .

$20 + 2 \sqrt{5} (3 \sqrt{3}) - 3 \sqrt{3} (2 \sqrt{5}) - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.4

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

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Step 2.1.4.1

Multiply $3$ by $2$ .

$20 + 6 \sqrt{5} \sqrt{3} - 3 \sqrt{3} (2 \sqrt{5}) - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.4.2

Combine using the product rule for radicals.

$20 + 6 \sqrt{3 \cdot 5} - 3 \sqrt{3} (2 \sqrt{5}) - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.4.3

Multiply $3$ by $5$ .

$20 + 6 \sqrt{15} - 3 \sqrt{3} (2 \sqrt{5}) - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.5

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

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Step 2.1.5.1

Multiply $2$ by $- 3$ .

$20 + 6 \sqrt{15} - 6 \sqrt{3} \sqrt{5} - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.5.2

Combine using the product rule for radicals.

$20 + 6 \sqrt{15} - 6 \sqrt{5 \cdot 3} - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.5.3

Multiply $5$ by $3$ .

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 3 \sqrt{3} (3 \sqrt{3})$

Step 2.1.6

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

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Step 2.1.6.1

Multiply $3$ by $- 3$ .

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 \sqrt{3} \sqrt{3}$

Step 2.1.6.2

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

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.1.6.3

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

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.1.6.4

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

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 {\sqrt{3}}^{1 + 1}$

Step 2.1.6.5

Add $1$ and $1$ .

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 {\sqrt{3}}^{2}$

Step 2.1.7

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

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Step 2.1.7.1

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

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 {(3^{\frac{1}{2}})}^{2}$

Step 2.1.7.2

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

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 2.1.7.3

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

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 \cdot 3^{\frac{2}{2}}$

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.

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 \cdot 3^{\frac{2}{2}}$

Step 2.1.7.4.2

Rewrite the expression.

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 \cdot 3^{1}$

Step 2.1.7.5

Evaluate the exponent.

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 9 \cdot 3$

Step 2.1.8

Multiply $- 9$ by $3$ .

$20 + 6 \sqrt{15} - 6 \sqrt{15} - 27$

Step 2.2

Subtract $27$ from $20$ .

$- 7 + 6 \sqrt{15} - 6 \sqrt{15}$

Step 2.3

Subtract $6 \sqrt{15}$ from $6 \sqrt{15}$ .

$- 7 + 0$

Step 2.4

Add $- 7$ and $0$ .

$- 7$