Linear Algebra Examples

$\frac{9 (266178) - 3986 \cdot 403}{\sqrt{9 (407404) - 15888196} \sqrt{9 (22487)} - (162409)}$

Step 1

Simplify the numerator.

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Step 1.1

Multiply $9$ by $266178$ .

$\frac{2395602 - 3986 \cdot 403}{\sqrt{9 (407404) - 15888196} \sqrt{9 (22487)} - (162409)}$

Step 1.2

Multiply $- 3986$ by $403$ .

$\frac{2395602 - 1606358}{\sqrt{9 (407404) - 15888196} \sqrt{9 (22487)} - (162409)}$

Step 1.3

Subtract $1606358$ from $2395602$ .

$\frac{789244}{\sqrt{9 (407404) - 15888196} \sqrt{9 (22487)} - (162409)}$

Step 2

Simplify the denominator.

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Step 2.1

Combine using the product rule for radicals.

$\frac{789244}{\sqrt{(9 (407404) - 15888196) (9 (22487))} - (162409)}$

Step 2.2

Multiply $22487$ by $9$ .

$\frac{789244}{\sqrt{(9 (407404) - 15888196) \cdot 202383} - (162409)}$

Step 2.3

Multiply $9$ by $407404$ .

$\frac{789244}{\sqrt{(3666636 - 15888196) \cdot 202383} - (162409)}$

Step 2.4

Subtract $15888196$ from $3666636$ .

$\frac{789244}{\sqrt{- 12221560 \cdot 202383} - (162409)}$

Step 2.5

Multiply $- 12221560$ by $202383$ .

$\frac{789244}{\sqrt{- 2473435977480} - (162409)}$

Step 2.6

Rewrite $- 2473435977480$ as $- 1 (2473435977480)$ .

$\frac{789244}{\sqrt{- 1 (2473435977480)} - (162409)}$

Step 2.7

Rewrite $\sqrt{- 1 (2473435977480)}$ as $\sqrt{- 1} \cdot \sqrt{2473435977480}$ .

$\frac{789244}{\sqrt{- 1} \cdot \sqrt{2473435977480} - (162409)}$

Step 2.8

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

$\frac{789244}{i \cdot \sqrt{2473435977480} - (162409)}$

Step 2.9

Rewrite $2473435977480$ as $6^{2} \cdot 68706554930$ .

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Step 2.9.1

Factor $36$ out of $2473435977480$ .

$\frac{789244}{i \cdot \sqrt{36 (68706554930)} - (162409)}$

Step 2.9.2

Rewrite $36$ as $6^{2}$ .

$\frac{789244}{i \cdot \sqrt{6^{2} \cdot 68706554930} - (162409)}$

Step 2.10

Pull terms out from under the radical.

$\frac{789244}{i \cdot (6 \sqrt{68706554930}) - (162409)}$

Step 2.11

Move $6$ to the left of $i$ .

$\frac{789244}{6 \cdot i \sqrt{68706554930} - (162409)}$

Step 2.12

Multiply $- 1$ by $162409$ .

$\frac{789244}{6 i \sqrt{68706554930} - 162409}$

Step 3

Multiply $\frac{789244}{6 i \sqrt{68706554930} - 162409}$ by $\frac{6 i \sqrt{68706554930} + 162409}{6 i \sqrt{68706554930} + 162409}$ .

$\frac{789244}{6 i \sqrt{68706554930} - 162409} \cdot \frac{6 i \sqrt{68706554930} + 162409}{6 i \sqrt{68706554930} + 162409}$

Step 4

Multiply $\frac{789244}{6 i \sqrt{68706554930} - 162409}$ by $\frac{6 i \sqrt{68706554930} + 162409}{6 i \sqrt{68706554930} + 162409}$ .

$\frac{789244 (6 i \sqrt{68706554930} + 162409)}{(6 i \sqrt{68706554930} - 162409) (6 i \sqrt{68706554930} + 162409)}$

Step 5

Expand the denominator using the FOIL method.

$\frac{789244 (6 i \sqrt{68706554930} + 162409)}{36 i^{2} {\sqrt{68706554930}}^{2} + 974454 i \sqrt{68706554930} - 974454 (i \sqrt{68706554930}) - 26376683281}$

Step 6

Simplify.

$\frac{789244 (6 i \sqrt{68706554930} + 162409)}{- 2499812660761}$

Step 7

Simplify the expression.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{789244 (6 i \sqrt{68706554930} + 162409)}{2499812660761}$

Step 7.2

Reorder $6 i \sqrt{68706554930}$ and $162409$ .

$- \frac{789244 (162409 + 6 i \sqrt{68706554930})}{2499812660761}$