Solve 1/-2sqrt{2+3}

$$\frac { 1 } { - 2 \sqrt { 2 } + 3 }$$

Evaluate

$$2\sqrt{2}+3\approx 5.828427125$$

Solution Steps

Rationalize the denominator of \frac{1}{-2\sqrt{2}+3} by multiplying numerator and denominator by -2\sqrt{2}-3.

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

Consider \left(-2\sqrt{2}+3\right)\left(-2\sqrt{2}-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(-2\sqrt{2}\right)^{2}.

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

Calculate -2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

$$\frac{-2\sqrt{2}-3}{4\times 2-3^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{-2\sqrt{2}-3}{8-3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{-2\sqrt{2}-3}{8-9}$$

Subtract 9 from 8 to get -1.

$$\frac{-2\sqrt{2}-3}{-1}$$

Anything divided by -1 gives its opposite. To find the opposite of -2\sqrt{2}-3, find the opposite of each term.

$$-\left(-2\sqrt{2}\right)-\left(-3\right)$$

The opposite of -2\sqrt{2} is 2\sqrt{2}.

$$2\sqrt{2}-\left(-3\right)$$

The opposite of -3 is 3.

$$2\sqrt{2}+3$$