Simplify (5-6 square root of -3)^2

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

Step 1

Simplify terms.

Simplify each term.

Rewrite $- 3$ as $- 1 (3)$ .

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

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Rewrite $\sqrt{- 1}$ as $i$ .

${(5 - 6 i \sqrt{3})}^{2}$

Rewrite ${(5 - 6 i \sqrt{3})}^{2}$ as $(5 - 6 i \sqrt{3}) (5 - 6 i \sqrt{3})$ .

$(5 - 6 i \sqrt{3}) (5 - 6 i \sqrt{3})$

Step 2

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

Apply the distributive property.

$5 (5 - 6 i \sqrt{3}) - 6 i \sqrt{3} (5 - 6 i \sqrt{3})$

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 (- 6 i \sqrt{3}) - 6 i \sqrt{3} \cdot 5 - 6 i \sqrt{3} (- 6 i \sqrt{3})$

Multiply $- 6$ by $5$ .

$25 - 30 (i \sqrt{3}) - 6 i \sqrt{3} \cdot 5 - 6 i \sqrt{3} (- 6 i \sqrt{3})$

Multiply $5$ by $- 6$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 6 i \sqrt{3} (- 6 i \sqrt{3})$

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

Multiply $- 6$ by $- 6$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 i \sqrt{3} (i \sqrt{3})$

Raise $i$ to the power of $1$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 (i^{1} i) \sqrt{3} \sqrt{3}$

Raise $i$ to the power of $1$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 (i^{1} i^{1}) \sqrt{3} \sqrt{3}$

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

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 i^{1 + 1} \sqrt{3} \sqrt{3}$

Add $1$ and $1$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 i^{2} \sqrt{3} \sqrt{3}$

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

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

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

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 i^{2} {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 i^{2} {\sqrt{3}}^{2}$

Rewrite $i^{2}$ as $- 1$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} + 36 \cdot - 1 {\sqrt{3}}^{2}$

Multiply $36$ by $- 1$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 36 {\sqrt{3}}^{2}$

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

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

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 36 {(3^{\frac{1}{2}})}^{2}$

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

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 36 \cdot 3^{\frac{1}{2} \cdot 2}$

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

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 36 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 36 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 36 \cdot 3^{1}$

Evaluate the exponent.

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 36 \cdot 3$

Multiply $- 36$ by $3$ .

$25 - 30 i \sqrt{3} - 30 i \sqrt{3} - 108$

Subtract $108$ from $25$ .

$- 30 i \sqrt{3} - 30 i \sqrt{3} - 83$

Subtract $30 i \sqrt{3}$ from $- 30 i \sqrt{3}$ .

$- 60 i \sqrt{3} - 83$

Step 4

Reorder $- 60 i \sqrt{3}$ and $- 83$ .

$- 83 - 60 i \sqrt{3}$

Simplify (5-6 square root of -3)^2

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