Solve frac{6/3sqrt{17+27}}{8}

$$\frac { \frac { 6 } { 3 \sqrt { 17 } + 27 } } { 8 }$$

Evaluate

$$\frac{9-\sqrt{17}}{256}\approx 0.019050369$$

Solution Steps

Express \frac{\frac{6}{3\sqrt{17}+27}}{8} as a single fraction.

$$\frac{6}{\left(3\sqrt{17}+27\right)\times 8}$$

Use the distributive property to multiply 3\sqrt{17}+27 by 8.

$$\frac{6}{24\sqrt{17}+216}$$

Rationalize the denominator of \frac{6}{24\sqrt{17}+216} by multiplying numerator and denominator by 24\sqrt{17}-216.

$$\frac{6\left(24\sqrt{17}-216\right)}{\left(24\sqrt{17}+216\right)\left(24\sqrt{17}-216\right)}$$

Consider \left(24\sqrt{17}+216\right)\left(24\sqrt{17}-216\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{6\left(24\sqrt{17}-216\right)}{\left(24\sqrt{17}\right)^{2}-216^{2}}$$

Expand \left(24\sqrt{17}\right)^{2}.

$$\frac{6\left(24\sqrt{17}-216\right)}{24^{2}\left(\sqrt{17}\right)^{2}-216^{2}}$$

Calculate 24 to the power of 2 and get 576.

$$\frac{6\left(24\sqrt{17}-216\right)}{576\left(\sqrt{17}\right)^{2}-216^{2}}$$

The square of \sqrt{17} is 17.

$$\frac{6\left(24\sqrt{17}-216\right)}{576\times 17-216^{2}}$$

Multiply 576 and 17 to get 9792.

$$\frac{6\left(24\sqrt{17}-216\right)}{9792-216^{2}}$$

Calculate 216 to the power of 2 and get 46656.

$$\frac{6\left(24\sqrt{17}-216\right)}{9792-46656}$$

Subtract 46656 from 9792 to get -36864.

$$\frac{6\left(24\sqrt{17}-216\right)}{-36864}$$

Divide 6\left(24\sqrt{17}-216\right) by -36864 to get -\frac{1}{6144}\left(24\sqrt{17}-216\right).

$$-\frac{1}{6144}\left(24\sqrt{17}-216\right)$$

Use the distributive property to multiply -\frac{1}{6144} by 24\sqrt{17}-216.

$$-\frac{1}{6144}\times 24\sqrt{17}-\frac{1}{6144}\left(-216\right)$$

Express -\frac{1}{6144}\times 24 as a single fraction.

$$\frac{-24}{6144}\sqrt{17}-\frac{1}{6144}\left(-216\right)$$

Reduce the fraction \frac{-24}{6144} to lowest terms by extracting and canceling out 24.

$$-\frac{1}{256}\sqrt{17}-\frac{1}{6144}\left(-216\right)$$

Express -\frac{1}{6144}\left(-216\right) as a single fraction.

$$-\frac{1}{256}\sqrt{17}+\frac{-\left(-216\right)}{6144}$$

Multiply -1 and -216 to get 216.

$$-\frac{1}{256}\sqrt{17}+\frac{216}{6144}$$

Reduce the fraction \frac{216}{6144} to lowest terms by extracting and canceling out 24.

$$-\frac{1}{256}\sqrt{17}+\frac{9}{256}$$