Solve 16/sqrt{2}-24/sqrt{32}-2sqrt{48}-3sqrt{8*54}

$$\frac{ 16 }{ \sqrt{ 2 } } - \frac{ 24 }{ \sqrt{ 32 } } -2 \sqrt{ 48 } -3 \sqrt{ 8 \cdot 54 }$$

Evaluate

$$5\sqrt{2}-44\sqrt{3}\approx -69.139167721$$

Solution Steps

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

$$\frac{16\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{24}{\sqrt{32}}-2\sqrt{48}-3\sqrt{8\times 54}$$

The square of \sqrt{2} is 2.

$$\frac{16\sqrt{2}}{2}-\frac{24}{\sqrt{32}}-2\sqrt{48}-3\sqrt{8\times 54}$$

Divide 16\sqrt{2} by 2 to get 8\sqrt{2}.

$$8\sqrt{2}-\frac{24}{\sqrt{32}}-2\sqrt{48}-3\sqrt{8\times 54}$$

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}.

$$8\sqrt{2}-\frac{24}{4\sqrt{2}}-2\sqrt{48}-3\sqrt{8\times 54}$$

Rationalize the denominator of \frac{24}{4\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$8\sqrt{2}-\frac{24\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}-2\sqrt{48}-3\sqrt{8\times 54}$$

The square of \sqrt{2} is 2.

$$8\sqrt{2}-\frac{24\sqrt{2}}{4\times 2}-2\sqrt{48}-3\sqrt{8\times 54}$$

Cancel out 2\times 4 in both numerator and denominator.

$$8\sqrt{2}-3\sqrt{2}-2\sqrt{48}-3\sqrt{8\times 54}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$8\sqrt{2}-3\sqrt{2}-2\times 4\sqrt{3}-3\sqrt{8\times 54}$$

Multiply -2 and 4 to get -8.

$$8\sqrt{2}-3\sqrt{2}-8\sqrt{3}-3\sqrt{8\times 54}$$

Multiply 8 and 54 to get 432.

$$8\sqrt{2}-3\sqrt{2}-8\sqrt{3}-3\sqrt{432}$$

Factor 432=12^{2}\times 3. Rewrite the square root of the product \sqrt{12^{2}\times 3} as the product of square roots \sqrt{12^{2}}\sqrt{3}. Take the square root of 12^{2}.

$$8\sqrt{2}-3\sqrt{2}-8\sqrt{3}-3\times 12\sqrt{3}$$

Multiply 3 and 12 to get 36.

$$8\sqrt{2}-3\sqrt{2}-8\sqrt{3}-36\sqrt{3}$$

Combine -8\sqrt{3} and -36\sqrt{3} to get -44\sqrt{3}.

$$8\sqrt{2}-3\sqrt{2}-44\sqrt{3}$$

Combine 8\sqrt{2} and -3\sqrt{2} to get 5\sqrt{2}.

$$5\sqrt{2}-44\sqrt{3}$$