Solve 16^2=(8sqrt{3})^2+n^2

$$16 ^ { 2 } = ( 8 \sqrt { 3 } ) ^ { 2 } + n ^ { 2 }$$

Solve for n

$$n=8$$

$$n=-8$$

Steps Using Difference of Squares

Calculate 16 to the power of 2 and get 256.

$$256=\left(8\sqrt{3}\right)^{2}+n^{2}$$

Expand \left(8\sqrt{3}\right)^{2}.

$$256=8^{2}\left(\sqrt{3}\right)^{2}+n^{2}$$

Calculate 8 to the power of 2 and get 64.

$$256=64\left(\sqrt{3}\right)^{2}+n^{2}$$

The square of \sqrt{3} is 3.

$$256=64\times 3+n^{2}$$

Multiply 64 and 3 to get 192.

$$256=192+n^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$192+n^{2}=256$$

Subtract 256 from both sides.

$$192+n^{2}-256=0$$

Subtract 256 from 192 to get -64.

$$-64+n^{2}=0$$

Consider -64+n^{2}. Rewrite -64+n^{2} as n^{2}-8^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(n-8\right)\left(n+8\right)=0$$

To find equation solutions, solve n-8=0 and n+8=0.

$$n=8 n=-8$$

Steps by Finding Square Root

Calculate 16 to the power of 2 and get 256.

$$256=\left(8\sqrt{3}\right)^{2}+n^{2}$$

Expand \left(8\sqrt{3}\right)^{2}.

$$256=8^{2}\left(\sqrt{3}\right)^{2}+n^{2}$$

Calculate 8 to the power of 2 and get 64.

$$256=64\left(\sqrt{3}\right)^{2}+n^{2}$$

The square of \sqrt{3} is 3.

$$256=64\times 3+n^{2}$$

Multiply 64 and 3 to get 192.

$$256=192+n^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$192+n^{2}=256$$

Subtract 192 from both sides.

$$n^{2}=256-192$$

Subtract 192 from 256 to get 64.

$$n^{2}=64$$

Take the square root of both sides of the equation.

$$n=8 n=-8$$

Steps Using the Quadratic Formula

Calculate 16 to the power of 2 and get 256.

$$256=\left(8\sqrt{3}\right)^{2}+n^{2}$$

Expand \left(8\sqrt{3}\right)^{2}.

$$256=8^{2}\left(\sqrt{3}\right)^{2}+n^{2}$$

Calculate 8 to the power of 2 and get 64.

$$256=64\left(\sqrt{3}\right)^{2}+n^{2}$$

The square of \sqrt{3} is 3.

$$256=64\times 3+n^{2}$$

Multiply 64 and 3 to get 192.

$$256=192+n^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$192+n^{2}=256$$

Subtract 256 from both sides.

$$192+n^{2}-256=0$$

Subtract 256 from 192 to get -64.

$$-64+n^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$n^{2}-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-64\right)}}{2}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-64\right)}}{2}$$

Multiply -4 times -64.

$$n=\frac{0±\sqrt{256}}{2}$$

Take the square root of 256.

$$n=\frac{0±16}{2}$$

Now solve the equation n=\frac{0±16}{2} when ± is plus. Divide 16 by 2.

$$n=8$$

Now solve the equation n=\frac{0±16}{2} when ± is minus. Divide -16 by 2.

$$n=-8$$

The equation is now solved.

$$n=8 n=-8$$