Solve frac{18sqrt{2}-1}{-6sqrt{2}+1}

$$\frac { 18 \sqrt { 2 } - 1 } { - 6 \sqrt { 2 } + 1 }$$

Evaluate

$$\frac{-12\sqrt{2}-215}{71}\approx -3.267191025$$

Solution Steps

Rationalize the denominator of \frac{18\sqrt{2}-1}{-6\sqrt{2}+1} by multiplying numerator and denominator by -6\sqrt{2}-1.

$$\frac{\left(18\sqrt{2}-1\right)\left(-6\sqrt{2}-1\right)}{\left(-6\sqrt{2}+1\right)\left(-6\sqrt{2}-1\right)}$$

Consider \left(-6\sqrt{2}+1\right)\left(-6\sqrt{2}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(18\sqrt{2}-1\right)\left(-6\sqrt{2}-1\right)}{\left(-6\sqrt{2}\right)^{2}-1^{2}}$$

Expand \left(-6\sqrt{2}\right)^{2}.

$$\frac{\left(18\sqrt{2}-1\right)\left(-6\sqrt{2}-1\right)}{\left(-6\right)^{2}\left(\sqrt{2}\right)^{2}-1^{2}}$$

Calculate -6 to the power of 2 and get 36.

$$\frac{\left(18\sqrt{2}-1\right)\left(-6\sqrt{2}-1\right)}{36\left(\sqrt{2}\right)^{2}-1^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(18\sqrt{2}-1\right)\left(-6\sqrt{2}-1\right)}{36\times 2-1^{2}}$$

Multiply 36 and 2 to get 72.

$$\frac{\left(18\sqrt{2}-1\right)\left(-6\sqrt{2}-1\right)}{72-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(18\sqrt{2}-1\right)\left(-6\sqrt{2}-1\right)}{72-1}$$

Subtract 1 from 72 to get 71.

$$\frac{\left(18\sqrt{2}-1\right)\left(-6\sqrt{2}-1\right)}{71}$$

Apply the distributive property by multiplying each term of 18\sqrt{2}-1 by each term of -6\sqrt{2}-1.

$$\frac{-108\left(\sqrt{2}\right)^{2}-18\sqrt{2}+6\sqrt{2}+1}{71}$$

The square of \sqrt{2} is 2.

$$\frac{-108\times 2-18\sqrt{2}+6\sqrt{2}+1}{71}$$

Multiply -108 and 2 to get -216.

$$\frac{-216-18\sqrt{2}+6\sqrt{2}+1}{71}$$

Combine -18\sqrt{2} and 6\sqrt{2} to get -12\sqrt{2}.

$$\frac{-216-12\sqrt{2}+1}{71}$$

Add -216 and 1 to get -215.

$$\frac{-215-12\sqrt{2}}{71}$$