Solve frac{sqrt{-48}+sqrt{-75}-sqrt{-147}}{sqrt{-12}}

$$\frac { \sqrt { - 48 } + \sqrt { - 75 } - \sqrt { - 147 } } { \sqrt { - 12 } }$$

Evaluate (complex solution)

$$1$$

Solution Steps

Factor -48=\left(4i\right)^{2}\times 3. Rewrite the square root of the product \sqrt{\left(4i\right)^{2}\times 3} as the product of square roots \sqrt{\left(4i\right)^{2}}\sqrt{3}. Take the square root of \left(4i\right)^{2}.

$$\frac{4i\sqrt{3}+\sqrt{-75}-\sqrt{-147}}{\sqrt{-12}}$$

Factor -75=\left(5i\right)^{2}\times 3. Rewrite the square root of the product \sqrt{\left(5i\right)^{2}\times 3} as the product of square roots \sqrt{\left(5i\right)^{2}}\sqrt{3}. Take the square root of \left(5i\right)^{2}.

$$\frac{4i\sqrt{3}+5i\sqrt{3}-\sqrt{-147}}{\sqrt{-12}}$$

Combine 4i\sqrt{3} and 5i\sqrt{3} to get 9i\sqrt{3}.

$$\frac{9i\sqrt{3}-\sqrt{-147}}{\sqrt{-12}}$$

Factor -147=\left(7i\right)^{2}\times 3. Rewrite the square root of the product \sqrt{\left(7i\right)^{2}\times 3} as the product of square roots \sqrt{\left(7i\right)^{2}}\sqrt{3}. Take the square root of \left(7i\right)^{2}.

$$\frac{9i\sqrt{3}-7i\sqrt{3}}{\sqrt{-12}}$$

Combine 9i\sqrt{3} and -7i\sqrt{3} to get 2i\sqrt{3}.

$$\frac{2i\sqrt{3}}{\sqrt{-12}}$$

Factor -12=\left(2i\right)^{2}\times 3. Rewrite the square root of the product \sqrt{\left(2i\right)^{2}\times 3} as the product of square roots \sqrt{\left(2i\right)^{2}}\sqrt{3}. Take the square root of \left(2i\right)^{2}.

$$\frac{2i\sqrt{3}}{2i\sqrt{3}}$$

Cancel out \sqrt{3} in both numerator and denominator.

$$\frac{2i}{2i}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{1}{\left(2i\right)^{0}}$$

Calculate 2i to the power of 0 and get 1.

$$\frac{1}{1}$$

Anything divided by one gives itself.

$$1$$

Real Part (complex solution)

$$1$$

Solution Steps

Factor -48=\left(4i\right)^{2}\times 3. Rewrite the square root of the product \sqrt{\left(4i\right)^{2}\times 3} as the product of square roots \sqrt{\left(4i\right)^{2}}\sqrt{3}. Take the square root of \left(4i\right)^{2}.

$$Re(\frac{4i\sqrt{3}+\sqrt{-75}-\sqrt{-147}}{\sqrt{-12}})$$

Factor -75=\left(5i\right)^{2}\times 3. Rewrite the square root of the product \sqrt{\left(5i\right)^{2}\times 3} as the product of square roots \sqrt{\left(5i\right)^{2}}\sqrt{3}. Take the square root of \left(5i\right)^{2}.

$$Re(\frac{4i\sqrt{3}+5i\sqrt{3}-\sqrt{-147}}{\sqrt{-12}})$$

Combine 4i\sqrt{3} and 5i\sqrt{3} to get 9i\sqrt{3}.

$$Re(\frac{9i\sqrt{3}-\sqrt{-147}}{\sqrt{-12}})$$

Factor -147=\left(7i\right)^{2}\times 3. Rewrite the square root of the product \sqrt{\left(7i\right)^{2}\times 3} as the product of square roots \sqrt{\left(7i\right)^{2}}\sqrt{3}. Take the square root of \left(7i\right)^{2}.

$$Re(\frac{9i\sqrt{3}-7i\sqrt{3}}{\sqrt{-12}})$$

Combine 9i\sqrt{3} and -7i\sqrt{3} to get 2i\sqrt{3}.

$$Re(\frac{2i\sqrt{3}}{\sqrt{-12}})$$

Factor -12=\left(2i\right)^{2}\times 3. Rewrite the square root of the product \sqrt{\left(2i\right)^{2}\times 3} as the product of square roots \sqrt{\left(2i\right)^{2}}\sqrt{3}. Take the square root of \left(2i\right)^{2}.

$$Re(\frac{2i\sqrt{3}}{2i\sqrt{3}})$$

Cancel out \sqrt{3} in both numerator and denominator.

$$Re(\frac{2i}{2i})$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$Re(\frac{1}{\left(2i\right)^{0}})$$

Calculate 2i to the power of 0 and get 1.

$$Re(\frac{1}{1})$$

Anything divided by one gives itself.

$$Re(1)$$

The real part of 1 is 1.

$$1$$

Evaluate

$$\text{Indeterminate}$$