Algebra Examples

$\sqrt{{(\sqrt{11} - 3)}^{2} + {(\sqrt{11} + 3)}^{2}}$

Step 1

Rewrite ${(\sqrt{11} - 3)}^{2}$ as $(\sqrt{11} - 3) (\sqrt{11} - 3)$ .

$\sqrt{(\sqrt{11} - 3) (\sqrt{11} - 3) + {(\sqrt{11} + 3)}^{2}}$

Step 2

Expand $(\sqrt{11} - 3) (\sqrt{11} - 3)$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

$\sqrt{\sqrt{11} (\sqrt{11} - 3) - 3 (\sqrt{11} - 3) + {(\sqrt{11} + 3)}^{2}}$

Step 2.2

Apply the distributive property.

$\sqrt{\sqrt{11} \sqrt{11} + \sqrt{11} \cdot - 3 - 3 (\sqrt{11} - 3) + {(\sqrt{11} + 3)}^{2}}$

Step 2.3

Apply the distributive property.

$\sqrt{\sqrt{11} \sqrt{11} + \sqrt{11} \cdot - 3 - 3 \sqrt{11} - 3 \cdot - 3 + {(\sqrt{11} + 3)}^{2}}$

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Combine using the product rule for radicals.

$\sqrt{\sqrt{11 \cdot 11} + \sqrt{11} \cdot - 3 - 3 \sqrt{11} - 3 \cdot - 3 + {(\sqrt{11} + 3)}^{2}}$

Step 3.1.2

Multiply $11$ by $11$ .

$\sqrt{\sqrt{121} + \sqrt{11} \cdot - 3 - 3 \sqrt{11} - 3 \cdot - 3 + {(\sqrt{11} + 3)}^{2}}$

Step 3.1.3

Rewrite $121$ as $11^{2}$ .

$\sqrt{\sqrt{11^{2}} + \sqrt{11} \cdot - 3 - 3 \sqrt{11} - 3 \cdot - 3 + {(\sqrt{11} + 3)}^{2}}$

Step 3.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{11 + \sqrt{11} \cdot - 3 - 3 \sqrt{11} - 3 \cdot - 3 + {(\sqrt{11} + 3)}^{2}}$

Step 3.1.5

Move $- 3$ to the left of $\sqrt{11}$ .

$\sqrt{11 - 3 \cdot \sqrt{11} - 3 \sqrt{11} - 3 \cdot - 3 + {(\sqrt{11} + 3)}^{2}}$

Step 3.1.6

Multiply $- 3$ by $- 3$ .

$\sqrt{11 - 3 \sqrt{11} - 3 \sqrt{11} + 9 + {(\sqrt{11} + 3)}^{2}}$

Step 3.2

Add $11$ and $9$ .

$\sqrt{20 - 3 \sqrt{11} - 3 \sqrt{11} + {(\sqrt{11} + 3)}^{2}}$

Step 3.3

Subtract $3 \sqrt{11}$ from $- 3 \sqrt{11}$ .

$\sqrt{20 - 6 \sqrt{11} + {(\sqrt{11} + 3)}^{2}}$

Step 4

Rewrite ${(\sqrt{11} + 3)}^{2}$ as $(\sqrt{11} + 3) (\sqrt{11} + 3)$ .

$\sqrt{20 - 6 \sqrt{11} + (\sqrt{11} + 3) (\sqrt{11} + 3)}$

Step 5

Expand $(\sqrt{11} + 3) (\sqrt{11} + 3)$ using the FOIL Method.

Tap for more steps...

Step 5.1

Apply the distributive property.

$\sqrt{20 - 6 \sqrt{11} + \sqrt{11} (\sqrt{11} + 3) + 3 (\sqrt{11} + 3)}$

Step 5.2

Apply the distributive property.

$\sqrt{20 - 6 \sqrt{11} + \sqrt{11} \sqrt{11} + \sqrt{11} \cdot 3 + 3 (\sqrt{11} + 3)}$

Step 5.3

Apply the distributive property.

$\sqrt{20 - 6 \sqrt{11} + \sqrt{11} \sqrt{11} + \sqrt{11} \cdot 3 + 3 \sqrt{11} + 3 \cdot 3}$

Step 6

Simplify and combine like terms.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Combine using the product rule for radicals.

$\sqrt{20 - 6 \sqrt{11} + \sqrt{11 \cdot 11} + \sqrt{11} \cdot 3 + 3 \sqrt{11} + 3 \cdot 3}$

Step 6.1.2

Multiply $11$ by $11$ .

$\sqrt{20 - 6 \sqrt{11} + \sqrt{121} + \sqrt{11} \cdot 3 + 3 \sqrt{11} + 3 \cdot 3}$

Step 6.1.3

Rewrite $121$ as $11^{2}$ .

$\sqrt{20 - 6 \sqrt{11} + \sqrt{11^{2}} + \sqrt{11} \cdot 3 + 3 \sqrt{11} + 3 \cdot 3}$

Step 6.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{20 - 6 \sqrt{11} + 11 + \sqrt{11} \cdot 3 + 3 \sqrt{11} + 3 \cdot 3}$

Step 6.1.5

Move $3$ to the left of $\sqrt{11}$ .

$\sqrt{20 - 6 \sqrt{11} + 11 + 3 \cdot \sqrt{11} + 3 \sqrt{11} + 3 \cdot 3}$

Step 6.1.6

Multiply $3$ by $3$ .

$\sqrt{20 - 6 \sqrt{11} + 11 + 3 \sqrt{11} + 3 \sqrt{11} + 9}$

Step 6.2

Add $11$ and $9$ .

$\sqrt{20 - 6 \sqrt{11} + 20 + 3 \sqrt{11} + 3 \sqrt{11}}$

Step 6.3

Add $3 \sqrt{11}$ and $3 \sqrt{11}$ .

$\sqrt{20 - 6 \sqrt{11} + 20 + 6 \sqrt{11}}$

Step 7

Add $20$ and $20$ .

$\sqrt{40 - 6 \sqrt{11} + 6 \sqrt{11}}$

Step 8

Add $- 6 \sqrt{11}$ and $6 \sqrt{11}$ .

$\sqrt{40 + 0}$

Step 9

Add $40$ and $0$ .

$\sqrt{40}$

Step 10

Rewrite $40$ as $2^{2} \cdot 10$ .

Tap for more steps...

Step 10.1

Factor $4$ out of $40$ .

$\sqrt{4 (10)}$

Step 10.2

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

$\sqrt{2^{2} \cdot 10}$

Step 11

Pull terms out from under the radical.

$2 \sqrt{10}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{10}$

Decimal Form:

$6.32455532 \dots$