Solve sqrt{18}-(sqrt{8}-sqrt{125})quadquadquad(4)quad1/2(sqrt{2}+sqrt{3})-3/4(sqrt{2}+sqrt{27})

$$\sqrt { 18 } - ( \sqrt { 8 } - \sqrt { 125 } ) \quad \quad \quad ( 4 ) \quad \frac { 1 } { 2 } ( \sqrt { 2 } + \sqrt { 3 } ) - \frac { 3 } { 4 } ( \sqrt { 2 } + \sqrt { 27 } )$$

Evaluate

$$\frac{9\sqrt{2}}{4}-\frac{9\sqrt{3}}{4}+10\sqrt{10}+10\sqrt{15}-4\sqrt{6}-8\approx 51.839517291$$

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$3\sqrt{2}-\left(\sqrt{8}-\sqrt{125}\right)\times 4\times \frac{1}{2}\left(\sqrt{2}+\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

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}.

$$3\sqrt{2}-\left(2\sqrt{2}-\sqrt{125}\right)\times 4\times \frac{1}{2}\left(\sqrt{2}+\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$3\sqrt{2}-\left(2\sqrt{2}-5\sqrt{5}\right)\times 4\times \frac{1}{2}\left(\sqrt{2}+\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

Multiply 4 and \frac{1}{2} to get \frac{4}{2}.

$$3\sqrt{2}-\left(2\sqrt{2}-5\sqrt{5}\right)\times \frac{4}{2}\left(\sqrt{2}+\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

Divide 4 by 2 to get 2.

$$3\sqrt{2}-\left(2\sqrt{2}-5\sqrt{5}\right)\times 2\left(\sqrt{2}+\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

Use the distributive property to multiply 2\sqrt{2}-5\sqrt{5} by 2.

$$3\sqrt{2}-\left(4\sqrt{2}-10\sqrt{5}\right)\left(\sqrt{2}+\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

Apply the distributive property by multiplying each term of 4\sqrt{2}-10\sqrt{5} by each term of \sqrt{2}+\sqrt{3}.

$$3\sqrt{2}-\left(4\left(\sqrt{2}\right)^{2}+4\sqrt{2}\sqrt{3}-10\sqrt{5}\sqrt{2}-10\sqrt{5}\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

The square of \sqrt{2} is 2.

$$3\sqrt{2}-\left(4\times 2+4\sqrt{2}\sqrt{3}-10\sqrt{5}\sqrt{2}-10\sqrt{5}\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

Multiply 4 and 2 to get 8.

$$3\sqrt{2}-\left(8+4\sqrt{2}\sqrt{3}-10\sqrt{5}\sqrt{2}-10\sqrt{5}\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

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

$$3\sqrt{2}-\left(8+4\sqrt{6}-10\sqrt{5}\sqrt{2}-10\sqrt{5}\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$3\sqrt{2}-\left(8+4\sqrt{6}-10\sqrt{10}-10\sqrt{5}\sqrt{3}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$3\sqrt{2}-\left(8+4\sqrt{6}-10\sqrt{10}-10\sqrt{15}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

To find the opposite of 8+4\sqrt{6}-10\sqrt{10}-10\sqrt{15}, find the opposite of each term.

$$3\sqrt{2}-8-4\sqrt{6}-\left(-10\sqrt{10}\right)-\left(-10\sqrt{15}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

The opposite of -10\sqrt{10} is 10\sqrt{10}.

$$3\sqrt{2}-8-4\sqrt{6}+10\sqrt{10}-\left(-10\sqrt{15}\right)-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

The opposite of -10\sqrt{15} is 10\sqrt{15}.

$$3\sqrt{2}-8-4\sqrt{6}+10\sqrt{10}+10\sqrt{15}-\frac{3}{4}\left(\sqrt{2}+\sqrt{27}\right)$$

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}.

$$3\sqrt{2}-8-4\sqrt{6}+10\sqrt{10}+10\sqrt{15}-\frac{3}{4}\left(\sqrt{2}+3\sqrt{3}\right)$$

Use the distributive property to multiply -\frac{3}{4} by \sqrt{2}+3\sqrt{3}.

$$3\sqrt{2}-8-4\sqrt{6}+10\sqrt{10}+10\sqrt{15}-\frac{3}{4}\sqrt{2}-\frac{3}{4}\times 3\sqrt{3}$$

Express -\frac{3}{4}\times 3 as a single fraction.

$$3\sqrt{2}-8-4\sqrt{6}+10\sqrt{10}+10\sqrt{15}-\frac{3}{4}\sqrt{2}+\frac{-3\times 3}{4}\sqrt{3}$$

Multiply -3 and 3 to get -9.

$$3\sqrt{2}-8-4\sqrt{6}+10\sqrt{10}+10\sqrt{15}-\frac{3}{4}\sqrt{2}+\frac{-9}{4}\sqrt{3}$$

Fraction \frac{-9}{4} can be rewritten as -\frac{9}{4} by extracting the negative sign.

$$3\sqrt{2}-8-4\sqrt{6}+10\sqrt{10}+10\sqrt{15}-\frac{3}{4}\sqrt{2}-\frac{9}{4}\sqrt{3}$$

Combine 3\sqrt{2} and -\frac{3}{4}\sqrt{2} to get \frac{9}{4}\sqrt{2}.

$$\frac{9}{4}\sqrt{2}-8-4\sqrt{6}+10\sqrt{10}+10\sqrt{15}-\frac{9}{4}\sqrt{3}$$

Factor

$$\frac{9 \sqrt{2} + 40 \sqrt{10} + 40 \sqrt{15} - 9 \sqrt{3} - 16 \sqrt{6} - 32}{4} = 51.83951729093474$$