Solve frac{6+3sqrt{6}}{4-3sqrt{6}}=frac{6+3sqrt{6}}{4-3sqrt{6}}*4/4

$$\frac { 6 + 3 \sqrt { 6 } } { 4 - 3 \sqrt { 6 } } = \frac { 6 + 3 \sqrt { 6 } } { 4 - 3 \sqrt { 6 } } \cdot \frac { 4 } { 4 }$$

Verify

Solution Steps

Divide 4 by 4 to get 1.

$$\frac{6+3\sqrt{6}}{4-3\sqrt{6}}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

Rationalize the denominator of \frac{6+3\sqrt{6}}{4-3\sqrt{6}} by multiplying numerator and denominator by 4+3\sqrt{6}.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{\left(4-3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

Consider \left(4-3\sqrt{6}\right)\left(4+3\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{4^{2}-\left(-3\sqrt{6}\right)^{2}}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-\left(-3\sqrt{6}\right)^{2}}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

Expand \left(-3\sqrt{6}\right)^{2}.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-\left(-3\right)^{2}\left(\sqrt{6}\right)^{2}}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

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

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-9\left(\sqrt{6}\right)^{2}}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

The square of \sqrt{6} is 6.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-9\times 6}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

Multiply 9 and 6 to get 54.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-54}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

Subtract 54 from 16 to get -38.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{6+3\sqrt{6}}{4-3\sqrt{6}}\times 1$$

Rationalize the denominator of \frac{6+3\sqrt{6}}{4-3\sqrt{6}} by multiplying numerator and denominator by 4+3\sqrt{6}.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{\left(4-3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}\times 1$$

Consider \left(4-3\sqrt{6}\right)\left(4+3\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{4^{2}-\left(-3\sqrt{6}\right)^{2}}\times 1$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-\left(-3\sqrt{6}\right)^{2}}\times 1$$

Expand \left(-3\sqrt{6}\right)^{2}.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-\left(-3\right)^{2}\left(\sqrt{6}\right)^{2}}\times 1$$

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

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-9\left(\sqrt{6}\right)^{2}}\times 1$$

The square of \sqrt{6} is 6.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-9\times 6}\times 1$$

Multiply 9 and 6 to get 54.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{16-54}\times 1$$

Subtract 54 from 16 to get -38.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}\times 1$$

Express \frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}\times 1 as a single fraction.

$$\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}$$

Apply the distributive property by multiplying each term of 6+3\sqrt{6} by each term of 4+3\sqrt{6}.

$$\frac{24+18\sqrt{6}+12\sqrt{6}+9\left(\sqrt{6}\right)^{2}}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}$$

Combine 18\sqrt{6} and 12\sqrt{6} to get 30\sqrt{6}.

$$\frac{24+30\sqrt{6}+9\left(\sqrt{6}\right)^{2}}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}$$

The square of \sqrt{6} is 6.

$$\frac{24+30\sqrt{6}+9\times 6}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}$$

Multiply 9 and 6 to get 54.

$$\frac{24+30\sqrt{6}+54}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}$$

Add 24 and 54 to get 78.

$$\frac{78+30\sqrt{6}}{-38}=\frac{\left(6+3\sqrt{6}\right)\left(4+3\sqrt{6}\right)}{-38}$$

Apply the distributive property by multiplying each term of 6+3\sqrt{6} by each term of 4+3\sqrt{6}.

$$\frac{78+30\sqrt{6}}{-38}=\frac{24+18\sqrt{6}+12\sqrt{6}+9\left(\sqrt{6}\right)^{2}}{-38}$$

Combine 18\sqrt{6} and 12\sqrt{6} to get 30\sqrt{6}.

$$\frac{78+30\sqrt{6}}{-38}=\frac{24+30\sqrt{6}+9\left(\sqrt{6}\right)^{2}}{-38}$$

The square of \sqrt{6} is 6.

$$\frac{78+30\sqrt{6}}{-38}=\frac{24+30\sqrt{6}+9\times 6}{-38}$$

Multiply 9 and 6 to get 54.

$$\frac{78+30\sqrt{6}}{-38}=\frac{24+30\sqrt{6}+54}{-38}$$

Add 24 and 54 to get 78.

$$\frac{78+30\sqrt{6}}{-38}=\frac{78+30\sqrt{6}}{-38}$$

Subtract \frac{78+30\sqrt{6}}{-38} from both sides.

$$\frac{78+30\sqrt{6}}{-38}-\frac{78+30\sqrt{6}}{-38}=0$$

Combine \frac{78+30\sqrt{6}}{-38} and -\frac{78+30\sqrt{6}}{-38} to get 0.

$$0=0$$

Compare 0 and 0.

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