Solve (sqrt{7}+2+2)(sqrt{7}+2)(sqrt{7}+2)*(4-sqrt{7})

$$( \sqrt { 7 } + 2 + 2 ) ( \sqrt { 7 } + 2 ) ( \sqrt { 7 } + 2 ) \cdot ( 4 - \sqrt { 7 } )$$

Evaluate

$$36\sqrt{7}+99\approx 194.247047198$$

Solution Steps

Multiply \sqrt{7}+2 and \sqrt{7}+2 to get \left(\sqrt{7}+2\right)^{2}.

$$\left(\sqrt{7}+2+2\right)\left(\sqrt{7}+2\right)^{2}\left(4-\sqrt{7}\right)$$

Add 2 and 2 to get 4.

$$\left(\sqrt{7}+4\right)\left(\sqrt{7}+2\right)^{2}\left(4-\sqrt{7}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{7}+2\right)^{2}.

$$\left(\sqrt{7}+4\right)\left(\left(\sqrt{7}\right)^{2}+4\sqrt{7}+4\right)\left(4-\sqrt{7}\right)$$

The square of \sqrt{7} is 7.

$$\left(\sqrt{7}+4\right)\left(7+4\sqrt{7}+4\right)\left(4-\sqrt{7}\right)$$

Add 7 and 4 to get 11.

$$\left(\sqrt{7}+4\right)\left(11+4\sqrt{7}\right)\left(4-\sqrt{7}\right)$$

Apply the distributive property by multiplying each term of \sqrt{7}+4 by each term of 11+4\sqrt{7}.

$$\left(11\sqrt{7}+4\left(\sqrt{7}\right)^{2}+44+16\sqrt{7}\right)\left(4-\sqrt{7}\right)$$

The square of \sqrt{7} is 7.

$$\left(11\sqrt{7}+4\times 7+44+16\sqrt{7}\right)\left(4-\sqrt{7}\right)$$

Multiply 4 and 7 to get 28.

$$\left(11\sqrt{7}+28+44+16\sqrt{7}\right)\left(4-\sqrt{7}\right)$$

Add 28 and 44 to get 72.

$$\left(11\sqrt{7}+72+16\sqrt{7}\right)\left(4-\sqrt{7}\right)$$

Combine 11\sqrt{7} and 16\sqrt{7} to get 27\sqrt{7}.

$$\left(27\sqrt{7}+72\right)\left(4-\sqrt{7}\right)$$

Apply the distributive property by multiplying each term of 27\sqrt{7}+72 by each term of 4-\sqrt{7}.

$$108\sqrt{7}-27\left(\sqrt{7}\right)^{2}+288-72\sqrt{7}$$

The square of \sqrt{7} is 7.

$$108\sqrt{7}-27\times 7+288-72\sqrt{7}$$

Multiply -27 and 7 to get -189.

$$108\sqrt{7}-189+288-72\sqrt{7}$$

Add -189 and 288 to get 99.

$$108\sqrt{7}+99-72\sqrt{7}$$

Combine 108\sqrt{7} and -72\sqrt{7} to get 36\sqrt{7}.

$$36\sqrt{7}+99$$

Factor

$$9 {(4 \sqrt{7} + 11)} = 194.247047198$$