Simplify (5- square root of -24)(5+ square root of -24)

$(5 - \sqrt{- 24}) (5 + \sqrt{- 24})$

Step 1

Rewrite $- 24$ as $- 1 (24)$ .

$(5 - \sqrt{- 1 (24)}) (5 + \sqrt{- 24})$

Step 2

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$(5 - (\sqrt{- 1} \cdot \sqrt{24})) (5 + \sqrt{- 24})$

Step 3

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

$(5 - (i \cdot \sqrt{24})) (5 + \sqrt{- 24})$

Step 4

Rewrite $- 24$ as $- 1 (24)$ .

$(5 - (i \cdot \sqrt{24})) (5 + \sqrt{- 1 (24)})$

Step 5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$(5 - (i \cdot \sqrt{24})) (5 + \sqrt{- 1} \cdot \sqrt{24})$

Step 6

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

$(5 - (i \cdot \sqrt{24})) (5 + i \cdot \sqrt{24})$

Step 7

Simplify terms.

Simplify each term.

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$(5 - (i \cdot \sqrt{4 (6)})) (5 + i \cdot \sqrt{24})$

Rewrite $4$ as $2^{2}$ .

$(5 - (i \cdot \sqrt{2^{2} \cdot 6})) (5 + i \cdot \sqrt{24})$

Pull terms out from under the radical.

$(5 - (i \cdot (2 \sqrt{6}))) (5 + i \cdot \sqrt{24})$

Move $2$ to the left of $i$ .

$(5 - (2 \cdot i \sqrt{6})) (5 + i \cdot \sqrt{24})$

Multiply $2$ by $- 1$ .

$(5 - 2 i \sqrt{6}) (5 + i \cdot \sqrt{24})$

Simplify each term.

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$(5 - 2 i \sqrt{6}) (5 + i \cdot \sqrt{4 (6)})$

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Move $2$ to the left of $i$ .

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

Step 8

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 9

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 (2 i \sqrt{6}) - 2 i \sqrt{6} \cdot 5 - 2 i \sqrt{6} (2 i \sqrt{6})$

Multiply $2$ by $5$ .

$25 + 10 (i \sqrt{6}) - 2 i \sqrt{6} \cdot 5 - 2 i \sqrt{6} (2 i \sqrt{6})$

Multiply $5$ by $- 2$ .

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 2 i \sqrt{6} (2 i \sqrt{6})$

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

Multiply $2$ by $- 2$ .

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 i \sqrt{6} (i \sqrt{6})$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 (i^{1} i) \sqrt{6} \sqrt{6}$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 (i^{1} i^{1}) \sqrt{6} \sqrt{6}$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 i^{1 + 1} \sqrt{6} \sqrt{6}$

Add $1$ and $1$ .

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 i^{2} \sqrt{6} \sqrt{6}$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 i^{2} ({\sqrt{6}}^{1} \sqrt{6})$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 i^{2} ({\sqrt{6}}^{1} {\sqrt{6}}^{1})$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 i^{2} {\sqrt{6}}^{1 + 1}$

Add $1$ and $1$ .

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 i^{2} {\sqrt{6}}^{2}$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} - 4 \cdot - 1 {\sqrt{6}}^{2}$

Multiply $- 4$ by $- 1$ .

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} + 4 {\sqrt{6}}^{2}$

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

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} + 4 {(6^{\frac{1}{2}})}^{2}$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} + 4 \cdot 6^{\frac{1}{2} \cdot 2}$

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

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} + 4 \cdot 6^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} + 4 \cdot 6^{\frac{2}{2}}$

Rewrite the expression.

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} + 4 \cdot 6^{1}$

Evaluate the exponent.

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} + 4 \cdot 6$

Multiply $4$ by $6$ .

$25 + 10 i \sqrt{6} - 10 i \sqrt{6} + 24$

Add $25$ and $24$ .

$10 i \sqrt{6} - 10 i \sqrt{6} + 49$

Subtract $10 i \sqrt{6}$ from $10 i \sqrt{6}$ .

$0 + 49$

Add $0$ and $49$ .

$49$

Simplify (5- square root of -24)(5+ square root of -24)

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