Solve 2sqrt{8}+1/2sqrt{18}-1/4sqrt{32}

$$2 \sqrt{ 8 } + \frac{ 1 }{ 2 } \sqrt{ 18 } - \frac{ 1 }{ 4 \sqrt{ 32 } }$$

Evaluate

$$\frac{175\sqrt{2}}{32}\approx 7.733980419$$

Solution Steps

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}.

$$2\times 2\sqrt{2}+\frac{1}{2}\sqrt{18}-\frac{1}{4\sqrt{32}}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{2}+\frac{1}{2}\sqrt{18}-\frac{1}{4\sqrt{32}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$4\sqrt{2}+\frac{1}{2}\times 3\sqrt{2}-\frac{1}{4\sqrt{32}}$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$4\sqrt{2}+\frac{3}{2}\sqrt{2}-\frac{1}{4\sqrt{32}}$$

Combine 4\sqrt{2} and \frac{3}{2}\sqrt{2} to get \frac{11}{2}\sqrt{2}.

$$\frac{11}{2}\sqrt{2}-\frac{1}{4\sqrt{32}}$$

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}.

$$\frac{11}{2}\sqrt{2}-\frac{1}{4\times 4\sqrt{2}}$$

Multiply 4 and 4 to get 16.

$$\frac{11}{2}\sqrt{2}-\frac{1}{16\sqrt{2}}$$

Rationalize the denominator of \frac{1}{16\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{11}{2}\sqrt{2}-\frac{\sqrt{2}}{16\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{11}{2}\sqrt{2}-\frac{\sqrt{2}}{16\times 2}$$

Multiply 16 and 2 to get 32.

$$\frac{11}{2}\sqrt{2}-\frac{\sqrt{2}}{32}$$

Combine \frac{11}{2}\sqrt{2} and -\frac{\sqrt{2}}{32} to get \frac{175}{32}\sqrt{2}.

$$\frac{175}{32}\sqrt{2}$$