Solve ((sqrt{32456})+1)^2

$$( ( \sqrt { 32456 } ) + 1 ) ^ { 2 }$$

Evaluate

$$4\sqrt{8114}+32457\approx 32817.310976796$$

Solution Steps

Factor 32456=2^{2}\times 8114. Rewrite the square root of the product \sqrt{2^{2}\times 8114} as the product of square roots \sqrt{2^{2}}\sqrt{8114}. Take the square root of 2^{2}.

$$\left(2\sqrt{8114}+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{8114}+1\right)^{2}.

$$4\left(\sqrt{8114}\right)^{2}+4\sqrt{8114}+1$$

The square of \sqrt{8114} is 8114.

$$4\times 8114+4\sqrt{8114}+1$$

Multiply 4 and 8114 to get 32456.

$$32456+4\sqrt{8114}+1$$

Add 32456 and 1 to get 32457.

$$32457+4\sqrt{8114}$$

Expand

$$4 \sqrt{8114} + 32457 = 32817.310976796$$

Solution Steps

Factor 32456=2^{2}\times 8114. Rewrite the square root of the product \sqrt{2^{2}\times 8114} as the product of square roots \sqrt{2^{2}}\sqrt{8114}. Take the square root of 2^{2}.

$$\left(2\sqrt{8114}+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{8114}+1\right)^{2}.

$$4\left(\sqrt{8114}\right)^{2}+4\sqrt{8114}+1$$

The square of \sqrt{8114} is 8114.

$$4\times 8114+4\sqrt{8114}+1$$

Multiply 4 and 8114 to get 32456.

$$32456+4\sqrt{8114}+1$$

Add 32456 and 1 to get 32457.

$$32457+4\sqrt{8114}$$