Solve 10/4sqrt{2-3sqrt{3}}

$$\frac{ 10 }{ 4 \sqrt{ 2 } -3 \sqrt{ 3 } }$$

Evaluate

$$2\left(3\sqrt{3}+4\sqrt{2}\right)\approx 21.706013344$$

Solution Steps

Rationalize the denominator of \frac{10}{4\sqrt{2}-3\sqrt{3}} by multiplying numerator and denominator by 4\sqrt{2}+3\sqrt{3}.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{\left(4\sqrt{2}-3\sqrt{3}\right)\left(4\sqrt{2}+3\sqrt{3}\right)}$$

Consider \left(4\sqrt{2}-3\sqrt{3}\right)\left(4\sqrt{2}+3\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}.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{\left(4\sqrt{2}\right)^{2}-\left(-3\sqrt{3}\right)^{2}}$$

Expand \left(4\sqrt{2}\right)^{2}.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{4^{2}\left(\sqrt{2}\right)^{2}-\left(-3\sqrt{3}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{16\left(\sqrt{2}\right)^{2}-\left(-3\sqrt{3}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{16\times 2-\left(-3\sqrt{3}\right)^{2}}$$

Multiply 16 and 2 to get 32.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{32-\left(-3\sqrt{3}\right)^{2}}$$

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

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{32-\left(-3\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{32-9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{32-9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{32-27}$$

Subtract 27 from 32 to get 5.

$$\frac{10\left(4\sqrt{2}+3\sqrt{3}\right)}{5}$$

Divide 10\left(4\sqrt{2}+3\sqrt{3}\right) by 5 to get 2\left(4\sqrt{2}+3\sqrt{3}\right).

$$2\left(4\sqrt{2}+3\sqrt{3}\right)$$

Use the distributive property to multiply 2 by 4\sqrt{2}+3\sqrt{3}.

$$8\sqrt{2}+6\sqrt{3}$$