Solve 5sqrt{12}+7sqrt{27}-sqrt{243}-1/2sqrt{75}

$$5 \sqrt { 12 } + 7 \sqrt { 27 } - \sqrt { 243 } - \frac { 1 } { 2 } \sqrt { 75 }$$

Evaluate

$$\frac{39\sqrt{3}}{2}\approx 33.774990748$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$5\times 2\sqrt{3}+7\sqrt{27}-\sqrt{243}-\frac{1}{2}\sqrt{75}$$

Multiply 5 and 2 to get 10.

$$10\sqrt{3}+7\sqrt{27}-\sqrt{243}-\frac{1}{2}\sqrt{75}$$

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}.

$$10\sqrt{3}+7\times 3\sqrt{3}-\sqrt{243}-\frac{1}{2}\sqrt{75}$$

Multiply 7 and 3 to get 21.

$$10\sqrt{3}+21\sqrt{3}-\sqrt{243}-\frac{1}{2}\sqrt{75}$$

Combine 10\sqrt{3} and 21\sqrt{3} to get 31\sqrt{3}.

$$31\sqrt{3}-\sqrt{243}-\frac{1}{2}\sqrt{75}$$

Factor 243=9^{2}\times 3. Rewrite the square root of the product \sqrt{9^{2}\times 3} as the product of square roots \sqrt{9^{2}}\sqrt{3}. Take the square root of 9^{2}.

$$31\sqrt{3}-9\sqrt{3}-\frac{1}{2}\sqrt{75}$$

Combine 31\sqrt{3} and -9\sqrt{3} to get 22\sqrt{3}.

$$22\sqrt{3}-\frac{1}{2}\sqrt{75}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$22\sqrt{3}-\frac{1}{2}\times 5\sqrt{3}$$

Express -\frac{1}{2}\times 5 as a single fraction.

$$22\sqrt{3}+\frac{-5}{2}\sqrt{3}$$

Fraction \frac{-5}{2} can be rewritten as -\frac{5}{2} by extracting the negative sign.

$$22\sqrt{3}-\frac{5}{2}\sqrt{3}$$

Combine 22\sqrt{3} and -\frac{5}{2}\sqrt{3} to get \frac{39}{2}\sqrt{3}.

$$\frac{39}{2}\sqrt{3}$$