Solve frac{21sqrt{15}}{512+5sqrt{3}}

$$\frac { 21 \sqrt { 15 } } { 512 + 5 \sqrt { 3 } }$$

Evaluate

$$\frac{10752\sqrt{15}-315\sqrt{5}}{262069}\approx 0.156210599$$

Solution Steps

Rationalize the denominator of \frac{21\sqrt{15}}{512+5\sqrt{3}} by multiplying numerator and denominator by 512-5\sqrt{3}.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{\left(512+5\sqrt{3}\right)\left(512-5\sqrt{3}\right)}$$

Consider \left(512+5\sqrt{3}\right)\left(512-5\sqrt{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}.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{512^{2}-\left(5\sqrt{3}\right)^{2}}$$

Calculate 512 to the power of 2 and get 262144.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-\left(5\sqrt{3}\right)^{2}}$$

Expand \left(5\sqrt{3}\right)^{2}.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-5^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-25\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-25\times 3}$$

Multiply 25 and 3 to get 75.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-75}$$

Subtract 75 from 262144 to get 262069.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262069}$$

Use the distributive property to multiply 21\sqrt{15} by 512-5\sqrt{3}.

$$\frac{10752\sqrt{15}-105\sqrt{3}\sqrt{15}}{262069}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\frac{10752\sqrt{15}-105\sqrt{3}\sqrt{3}\sqrt{5}}{262069}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{10752\sqrt{15}-105\times 3\sqrt{5}}{262069}$$

Multiply -105 and 3 to get -315.

$$\frac{10752\sqrt{15}-315\sqrt{5}}{262069}$$