Solve 4sqrt{15/8}text{i}1/5sqrt{750}

$$4 \sqrt { \frac { 15 } { 8 } } \text { i } \frac { 1 } { 5 } \sqrt { 750 }$$

Evaluate

$$30i$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{15}{8}} as the division of square roots \frac{\sqrt{15}}{\sqrt{8}}.

$$4i\times \frac{\sqrt{15}}{\sqrt{8}}\times \frac{1}{5}\sqrt{750}$$

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}.

$$4i\times \frac{\sqrt{15}}{2\sqrt{2}}\times \frac{1}{5}\sqrt{750}$$

Rationalize the denominator of \frac{\sqrt{15}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$4i\times \frac{\sqrt{15}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}\times \frac{1}{5}\sqrt{750}$$

The square of \sqrt{2} is 2.

$$4i\times \frac{\sqrt{15}\sqrt{2}}{2\times 2}\times \frac{1}{5}\sqrt{750}$$

To multiply \sqrt{15} and \sqrt{2}, multiply the numbers under the square root.

$$4i\times \frac{\sqrt{30}}{2\times 2}\times \frac{1}{5}\sqrt{750}$$

Multiply 2 and 2 to get 4.

$$4i\times \frac{\sqrt{30}}{4}\times \frac{1}{5}\sqrt{750}$$

Multiply 4i and \frac{1}{5} to get \frac{4}{5}i.

$$\frac{4}{5}i\times \frac{\sqrt{30}}{4}\sqrt{750}$$

Factor 750=5^{2}\times 30. Rewrite the square root of the product \sqrt{5^{2}\times 30} as the product of square roots \sqrt{5^{2}}\sqrt{30}. Take the square root of 5^{2}.

$$\frac{4}{5}i\times \frac{\sqrt{30}}{4}\times 5\sqrt{30}$$

Multiply \frac{4}{5}i and 5 to get 4i.

$$4i\times \frac{\sqrt{30}}{4}\sqrt{30}$$

Express \frac{\sqrt{30}}{4}\sqrt{30} as a single fraction.

$$4i\times \frac{\sqrt{30}\sqrt{30}}{4}$$

Multiply \sqrt{30} and \sqrt{30} to get 30.

$$4i\times \frac{30}{4}$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$4i\times \frac{15}{2}$$

Multiply 4i and \frac{15}{2} to get 30i.

$$30i$$

Real Part

$$0$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{15}{8}} as the division of square roots \frac{\sqrt{15}}{\sqrt{8}}.

$$Re(4i\times \frac{\sqrt{15}}{\sqrt{8}}\times \frac{1}{5}\sqrt{750})$$

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}.

$$Re(4i\times \frac{\sqrt{15}}{2\sqrt{2}}\times \frac{1}{5}\sqrt{750})$$

Rationalize the denominator of \frac{\sqrt{15}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$Re(4i\times \frac{\sqrt{15}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}\times \frac{1}{5}\sqrt{750})$$

The square of \sqrt{2} is 2.

$$Re(4i\times \frac{\sqrt{15}\sqrt{2}}{2\times 2}\times \frac{1}{5}\sqrt{750})$$

To multiply \sqrt{15} and \sqrt{2}, multiply the numbers under the square root.

$$Re(4i\times \frac{\sqrt{30}}{2\times 2}\times \frac{1}{5}\sqrt{750})$$

Multiply 2 and 2 to get 4.

$$Re(4i\times \frac{\sqrt{30}}{4}\times \frac{1}{5}\sqrt{750})$$

Multiply 4i and \frac{1}{5} to get \frac{4}{5}i.

$$Re(\frac{4}{5}i\times \frac{\sqrt{30}}{4}\sqrt{750})$$

Factor 750=5^{2}\times 30. Rewrite the square root of the product \sqrt{5^{2}\times 30} as the product of square roots \sqrt{5^{2}}\sqrt{30}. Take the square root of 5^{2}.

$$Re(\frac{4}{5}i\times \frac{\sqrt{30}}{4}\times 5\sqrt{30})$$

Multiply \frac{4}{5}i and 5 to get 4i.

$$Re(4i\times \frac{\sqrt{30}}{4}\sqrt{30})$$

Express \frac{\sqrt{30}}{4}\sqrt{30} as a single fraction.

$$Re(4i\times \frac{\sqrt{30}\sqrt{30}}{4})$$

Multiply \sqrt{30} and \sqrt{30} to get 30.

$$Re(4i\times \frac{30}{4})$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$Re(4i\times \frac{15}{2})$$

Multiply 4i and \frac{15}{2} to get 30i.

$$Re(30i)$$

The real part of 30i is 0.

$$0$$