Solve frac{3sqrt{32}+2sqrt{16}}{2sqrt{8}}

$$\frac { 3 \sqrt { 32 } + 2 \sqrt { 16 } } { 2 \sqrt { 8 } }$$

Evaluate

$$\sqrt{2}+3\approx 4.414213562$$

Solution Steps

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\frac{3\times 4\sqrt{2}+2\sqrt{16}}{2\sqrt{8}}$$

Multiply 3 and 4 to get 12.

$$\frac{12\sqrt{2}+2\sqrt{16}}{2\sqrt{8}}$$

Calculate the square root of 16 and get 4.

$$\frac{12\sqrt{2}+2\times 4}{2\sqrt{8}}$$

Multiply 2 and 4 to get 8.

$$\frac{12\sqrt{2}+8}{2\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{12\sqrt{2}+8}{2\times 2\sqrt{2}}$$

Multiply 2 and 2 to get 4.

$$\frac{12\sqrt{2}+8}{4\sqrt{2}}$$

Rationalize the denominator of \frac{12\sqrt{2}+8}{4\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(12\sqrt{2}+8\right)\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(12\sqrt{2}+8\right)\sqrt{2}}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(12\sqrt{2}+8\right)\sqrt{2}}{8}$$

Use the distributive property to multiply 12\sqrt{2}+8 by \sqrt{2}.

$$\frac{12\left(\sqrt{2}\right)^{2}+8\sqrt{2}}{8}$$

The square of \sqrt{2} is 2.

$$\frac{12\times 2+8\sqrt{2}}{8}$$

Multiply 12 and 2 to get 24.

$$\frac{24+8\sqrt{2}}{8}$$

Divide each term of 24+8\sqrt{2} by 8 to get 3+\sqrt{2}.

$$3+\sqrt{2}$$