Solve 3sqrt{12}-5sqrt{15}-2sqrt{27}sqrt{32}+5sqrt{12}

$$3 \sqrt{ 12 } -5 \sqrt{ 15 } -2 \sqrt{ 27 } \sqrt{ 32 } +5 \sqrt{ 12 }$$

Evaluate

$$16\sqrt{3}-5\sqrt{15}-24\sqrt{6}\approx -50.439857637$$

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}.

$$3\times 2\sqrt{3}-5\sqrt{15}-2\sqrt{27}\sqrt{32}+5\sqrt{12}$$

Multiply 3 and 2 to get 6.

$$6\sqrt{3}-5\sqrt{15}-2\sqrt{27}\sqrt{32}+5\sqrt{12}$$

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}.

$$6\sqrt{3}-5\sqrt{15}-2\times 3\sqrt{3}\sqrt{32}+5\sqrt{12}$$

Multiply 2 and 3 to get 6.

$$6\sqrt{3}-5\sqrt{15}-6\sqrt{3}\sqrt{32}+5\sqrt{12}$$

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}.

$$6\sqrt{3}-5\sqrt{15}-6\sqrt{3}\times 4\sqrt{2}+5\sqrt{12}$$

Multiply 6 and 4 to get 24.

$$6\sqrt{3}-5\sqrt{15}-24\sqrt{3}\sqrt{2}+5\sqrt{12}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$6\sqrt{3}-5\sqrt{15}-24\sqrt{6}+5\sqrt{12}$$

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}.

$$6\sqrt{3}-5\sqrt{15}-24\sqrt{6}+5\times 2\sqrt{3}$$

Multiply 5 and 2 to get 10.

$$6\sqrt{3}-5\sqrt{15}-24\sqrt{6}+10\sqrt{3}$$

Combine 6\sqrt{3} and 10\sqrt{3} to get 16\sqrt{3}.

$$16\sqrt{3}-5\sqrt{15}-24\sqrt{6}$$