Solve [1-(1-3/2)^-2sqrt{1-3/4}+3/2]^-1

$$[ 1 - ( 1 - \frac { 3 } { 2 } ) ^ { - 2 } \sqrt { 1 - \frac { 3 } { 4 } } + \frac { 3 } { 2 } ] ^ { - 1 }$$

Evaluate

$$2$$

Solution Steps

Subtract \frac{3}{2} from 1 to get -\frac{1}{2}.

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

Calculate -\frac{1}{2} to the power of -2 and get 4.

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

Subtract \frac{3}{4} from 1 to get \frac{1}{4}.

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

Rewrite the square root of the division \frac{1}{4} as the division of square roots \frac{\sqrt{1}}{\sqrt{4}}. Take the square root of both numerator and denominator.

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

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

$$\left(1-2+\frac{3}{2}\right)^{-1}$$

Subtract 2 from 1 to get -1.

$$\left(-1+\frac{3}{2}\right)^{-1}$$

Add -1 and \frac{3}{2} to get \frac{1}{2}.

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

Calculate \frac{1}{2} to the power of -1 and get 2.

$$2$$

Factor

$$2$$