Solve (4+sqrt{3})^2-(4-sqrt{3})^2

$$( 4 + \sqrt { 3 } ) ^ { 2 } - ( 4 - \sqrt { 3 } ) ^ { 2 }$$

Evaluate

$$16\sqrt{3}\approx 27.712812921$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+\sqrt{3}\right)^{2}.

$$16+8\sqrt{3}+\left(\sqrt{3}\right)^{2}-\left(4-\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$16+8\sqrt{3}+3-\left(4-\sqrt{3}\right)^{2}$$

Add 16 and 3 to get 19.

$$19+8\sqrt{3}-\left(4-\sqrt{3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{3}\right)^{2}.

$$19+8\sqrt{3}-\left(16-8\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$19+8\sqrt{3}-\left(16-8\sqrt{3}+3\right)$$

Add 16 and 3 to get 19.

$$19+8\sqrt{3}-\left(19-8\sqrt{3}\right)$$

To find the opposite of 19-8\sqrt{3}, find the opposite of each term.

$$19+8\sqrt{3}-19+8\sqrt{3}$$

Subtract 19 from 19 to get 0.

$$8\sqrt{3}+8\sqrt{3}$$

Combine 8\sqrt{3} and 8\sqrt{3} to get 16\sqrt{3}.

$$16\sqrt{3}$$

Expand

$$16 \sqrt{3} = 27.712812921$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+\sqrt{3}\right)^{2}.

$$16+8\sqrt{3}+\left(\sqrt{3}\right)^{2}-\left(4-\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$16+8\sqrt{3}+3-\left(4-\sqrt{3}\right)^{2}$$

Add 16 and 3 to get 19.

$$19+8\sqrt{3}-\left(4-\sqrt{3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{3}\right)^{2}.

$$19+8\sqrt{3}-\left(16-8\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$19+8\sqrt{3}-\left(16-8\sqrt{3}+3\right)$$

Add 16 and 3 to get 19.

$$19+8\sqrt{3}-\left(19-8\sqrt{3}\right)$$

To find the opposite of 19-8\sqrt{3}, find the opposite of each term.

$$19+8\sqrt{3}-19+8\sqrt{3}$$

Subtract 19 from 19 to get 0.

$$8\sqrt{3}+8\sqrt{3}$$

Combine 8\sqrt{3} and 8\sqrt{3} to get 16\sqrt{3}.

$$16\sqrt{3}$$