Solve 3sqrt{27}-3sqrt{1/64}sqrt{4}+3sqrt{8}

$$3 \sqrt{ 27 } -3 \sqrt{ \frac{ 1 }{ 64 } } \sqrt{ 4 } +3 \sqrt{ 8 }$$

Evaluate

$$6\sqrt{2}+9\sqrt{3}-\frac{3}{4}\approx 23.323738642$$

Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$3\times 3\sqrt{3}-3\sqrt{\frac{1}{64}}\sqrt{4}+3\sqrt{8}$$

Multiply 3 and 3 to get 9.

$$9\sqrt{3}-3\sqrt{\frac{1}{64}}\sqrt{4}+3\sqrt{8}$$

Rewrite the square root of the division \frac{1}{64} as the division of square roots \frac{\sqrt{1}}{\sqrt{64}}. Take the square root of both numerator and denominator.

$$9\sqrt{3}-3\times \frac{1}{8}\sqrt{4}+3\sqrt{8}$$

Multiply 3 and \frac{1}{8} to get \frac{3}{8}.

$$9\sqrt{3}-\frac{3}{8}\sqrt{4}+3\sqrt{8}$$

Calculate the square root of 4 and get 2.

$$9\sqrt{3}-\frac{3}{8}\times 2+3\sqrt{8}$$

Express \frac{3}{8}\times 2 as a single fraction.

$$9\sqrt{3}-\frac{3\times 2}{8}+3\sqrt{8}$$

Multiply 3 and 2 to get 6.

$$9\sqrt{3}-\frac{6}{8}+3\sqrt{8}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

$$9\sqrt{3}-\frac{3}{4}+3\sqrt{8}$$

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}.

$$9\sqrt{3}-\frac{3}{4}+3\times 2\sqrt{2}$$

Multiply 3 and 2 to get 6.

$$9\sqrt{3}-\frac{3}{4}+6\sqrt{2}$$

Factor

$$\frac{3 {(8 \sqrt{2} + 12 \sqrt{3} - 1)}}{4} = 23.323738642358464$$