Solve (7sqrt{2}+3sqrt{8}-5sqrt{50})divsqrt{2}

$$( 7 \sqrt { 2 } + 3 \sqrt { 8 } - 5 \sqrt { 50 } ) \div \sqrt { 2 }$$

Evaluate

$$-12$$

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}.

$$\frac{7\sqrt{2}+3\times 2\sqrt{2}-5\sqrt{50}}{\sqrt{2}}$$

Multiply 3 and 2 to get 6.

$$\frac{7\sqrt{2}+6\sqrt{2}-5\sqrt{50}}{\sqrt{2}}$$

Combine 7\sqrt{2} and 6\sqrt{2} to get 13\sqrt{2}.

$$\frac{13\sqrt{2}-5\sqrt{50}}{\sqrt{2}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\frac{13\sqrt{2}-5\times 5\sqrt{2}}{\sqrt{2}}$$

Multiply -5 and 5 to get -25.

$$\frac{13\sqrt{2}-25\sqrt{2}}{\sqrt{2}}$$

Combine 13\sqrt{2} and -25\sqrt{2} to get -12\sqrt{2}.

$$\frac{-12\sqrt{2}}{\sqrt{2}}$$

Rationalize the denominator of \frac{-12\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{-12\sqrt{2}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-12\sqrt{2}\sqrt{2}}{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{-12\times 2}{2}$$

Multiply -12 and 2 to get -24.

$$\frac{-24}{2}$$

Divide -24 by 2 to get -12.

$$-12$$

Factor

$$-12$$