Solve (sqrt{15}+2)^2(sqrt{15}-2)^2

$$( \sqrt { 15 } + 2 ) ^ { 2 } ( \sqrt { 15 } - 2 ) ^ { 2 }$$

Evaluate

$$121$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{15}+2\right)^{2}.

$$\left(\left(\sqrt{15}\right)^{2}+4\sqrt{15}+4\right)\left(\sqrt{15}-2\right)^{2}$$

The square of \sqrt{15} is 15.

$$\left(15+4\sqrt{15}+4\right)\left(\sqrt{15}-2\right)^{2}$$

Add 15 and 4 to get 19.

$$\left(19+4\sqrt{15}\right)\left(\sqrt{15}-2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{15}-2\right)^{2}.

$$\left(19+4\sqrt{15}\right)\left(\left(\sqrt{15}\right)^{2}-4\sqrt{15}+4\right)$$

The square of \sqrt{15} is 15.

$$\left(19+4\sqrt{15}\right)\left(15-4\sqrt{15}+4\right)$$

Add 15 and 4 to get 19.

$$\left(19+4\sqrt{15}\right)\left(19-4\sqrt{15}\right)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 19.

$$361-\left(4\sqrt{15}\right)^{2}$$

Expand \left(4\sqrt{15}\right)^{2}.

$$361-4^{2}\left(\sqrt{15}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$361-16\left(\sqrt{15}\right)^{2}$$

The square of \sqrt{15} is 15.

$$361-16\times 15$$

Multiply 16 and 15 to get 240.

$$361-240$$

Subtract 240 from 361 to get 121.

$$121$$

Factor

$$11^{2}$$