Solve {left(2sqrt{3}-3right)}^2-13/4+sqrt{3}-{left(sqrt{4}-3right)}^0

$${ \left(2 \sqrt{ 3 } -3 \right) }^{ 2 } - \frac{ 13 }{ 4+ \sqrt{ 3 } } - { \left( \sqrt{ 4 } -3 \right) }^{ 0 }$$

Evaluate

$$16-11\sqrt{3}\approx -3.052558883$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2\sqrt{3}-3\right)^{2}.

$$4\left(\sqrt{3}\right)^{2}-12\sqrt{3}+9-\frac{13}{4+\sqrt{3}}-\left(\sqrt{4}-3\right)^{0}$$

The square of \sqrt{3} is 3.

$$4\times 3-12\sqrt{3}+9-\frac{13}{4+\sqrt{3}}-\left(\sqrt{4}-3\right)^{0}$$

Multiply 4 and 3 to get 12.

$$12-12\sqrt{3}+9-\frac{13}{4+\sqrt{3}}-\left(\sqrt{4}-3\right)^{0}$$

Add 12 and 9 to get 21.

$$21-12\sqrt{3}-\frac{13}{4+\sqrt{3}}-\left(\sqrt{4}-3\right)^{0}$$

Rationalize the denominator of \frac{13}{4+\sqrt{3}} by multiplying numerator and denominator by 4-\sqrt{3}.

$$21-12\sqrt{3}-\frac{13\left(4-\sqrt{3}\right)}{\left(4+\sqrt{3}\right)\left(4-\sqrt{3}\right)}-\left(\sqrt{4}-3\right)^{0}$$

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

$$21-12\sqrt{3}-\frac{13\left(4-\sqrt{3}\right)}{4^{2}-\left(\sqrt{3}\right)^{2}}-\left(\sqrt{4}-3\right)^{0}$$

Square 4. Square \sqrt{3}.

$$21-12\sqrt{3}-\frac{13\left(4-\sqrt{3}\right)}{16-3}-\left(\sqrt{4}-3\right)^{0}$$

Subtract 3 from 16 to get 13.

$$21-12\sqrt{3}-\frac{13\left(4-\sqrt{3}\right)}{13}-\left(\sqrt{4}-3\right)^{0}$$

Cancel out 13 and 13.

$$21-12\sqrt{3}-\left(4-\sqrt{3}\right)-\left(\sqrt{4}-3\right)^{0}$$

To find the opposite of 4-\sqrt{3}, find the opposite of each term.

$$21-12\sqrt{3}-4+\sqrt{3}-\left(\sqrt{4}-3\right)^{0}$$

Subtract 4 from 21 to get 17.

$$17-12\sqrt{3}+\sqrt{3}-\left(\sqrt{4}-3\right)^{0}$$

Combine -12\sqrt{3} and \sqrt{3} to get -11\sqrt{3}.

$$17-11\sqrt{3}-\left(\sqrt{4}-3\right)^{0}$$

Calculate the square root of 4 and get 2.

$$17-11\sqrt{3}-\left(2-3\right)^{0}$$

Subtract 3 from 2 to get -1.

$$17-11\sqrt{3}-\left(-1\right)^{0}$$

Calculate -1 to the power of 0 and get 1.

$$17-11\sqrt{3}-1$$

Subtract 1 from 17 to get 16.

$$16-11\sqrt{3}$$