g) Rationalize (2)/(3sqrt(5)+4)

Here are the solutions to the surd rationalization problems:

g) Rationalize $\frac{2}{3\sqrt{5}+4}$

• Step 1: Identify the conjugate of the denominator. The denominator is $3\sqrt{5}+4$. Its conjugate is $3\sqrt{5}-4$.

• Step 2: Multiply the numerator and denominator by the conjugate.

  $$\frac{2}{3\sqrt{5}+4} \times \frac{3\sqrt{5}-4}{3\sqrt{5}-4}$$

• Step 3: Simplify the numerator.

  $$2(3\sqrt{5}-4) = 6\sqrt{5} - 8$$

• Step 4: Simplify the denominator using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$. Here, $a = 3\sqrt{5}$ and $b = 4$.

  $$(3\sqrt{5})^2 - (4)^2 = (3^2 \times (\sqrt{5})^2) - 16 = (9 \times 5) - 16 = 45 - 16 = 29$$

• Step 5: Write the rationalized expression.

  $$\frac{6\sqrt{5}-8}{29}$$

  The rationalized form is $\boxed{\frac{6\sqrt{5}-8}{29}}$.

h) Rationalize $\frac{6}{2\sqrt{2}-1}$

• Step 1: Identify the conjugate of the denominator. The denominator is $2\sqrt{2}-1$. Its conjugate is $2\sqrt{2}+1$.

• Step 2: Multiply the numerator and denominator by the conjugate.

  $$\frac{6}{2\sqrt{2}-1} \times \frac{2\sqrt{2}+1}{2\sqrt{2}+1}$$

• Step 3: Simplify the numerator.

  $$6(2\sqrt{2}+1) = 12\sqrt{2} + 6$$

• Step 4: Simplify the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$. Here, $a = 2\sqrt{2}$ and $b = 1$.

  $$(2\sqrt{2})^2 - (1)^2 = (2^2 \times (\sqrt{2})^2) - 1 = (4 \times 2) - 1 = 8 - 1 = 7$$

• Step 5: Write the rationalized expression.

  $$\frac{12\sqrt{2}+6}{7}$$

  The rationalized form is $\boxed{\frac{12\sqrt{2}+6}{7}}$.

i) Rationalize $\frac{2\sqrt{3}+2}{\sqrt{3}-2}$

• Step 1: Identify the conjugate of the denominator. The denominator is $\sqrt{3}-2$. Its conjugate is $\sqrt{3}+2$.

• Step 2: Multiply the numerator and denominator by the conjugate.

  $$\frac{2\sqrt{3}+2}{\sqrt{3}-2} \times \frac{\sqrt{3}+2}{\sqrt{3}+2}$$

• Step 3: Simplify the numerator using FOIL (First, Outer, Inner, Last).

  $$(2\sqrt{3}+2)(\sqrt{3}+2) = (2\sqrt{3})(\sqrt{3}) + (2\sqrt{3})(2) + (2)(\sqrt{3}) + (2)(2)$$ $$= (2 \times 3) + 4\sqrt{3} + 2\sqrt{3} + 4$$ $$= 6 + 6\sqrt{3} + 4 = 10 + 6\sqrt{3}$$

• Step 4: Simplify the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$. Here, $a = \sqrt{3}$ and $b = 2$.

  $$(\sqrt{3})^2 - (2)^2 = 3 - 4 = -1$$

• Step 5: Write the rationalized expression.

  $$\frac{10+6\sqrt{3}}{-1} = -(10+6\sqrt{3}) = -10-6\sqrt{3}$$

  The rationalized form is $\boxed{-10-6\sqrt{3}}$.

j) Rationalize $\frac{2+\sqrt{5}}{2\sqrt{5}-1}$

• Step 1: Identify the conjugate of the denominator. The denominator is $2\sqrt{5}-1$. Its conjugate is $2\sqrt{5}+1$.

• Step 2: Multiply the numerator and denominator by the conjugate.

  $$\frac{2+\sqrt{5}}{2\sqrt{5}-1} \times \frac{2\sqrt{5}+1}{2\sqrt{5}+1}$$

• Step 3: Simplify the numerator using FOIL (First, Outer, Inner, Last).

  $$(2+\sqrt{5})(2\sqrt{5}+1) = (2)(2\sqrt{5}) + (2)(1) + (\sqrt{5})(2\sqrt{5}) + (\sqrt{5})(1)$$ $$= 4\sqrt{5} + 2 + (2 \times 5) + \sqrt{5}$$ $$= 4\sqrt{5} + 2 + 10 + \sqrt{5} = 12 + 5\sqrt{5}$$

• Step 4: Simplify the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$. Here, $a = 2\sqrt{5}$ and $b = 1$.

  $$(2\sqrt{5})^2 - (1)^2 = (2^2 \times (\sqrt{5})^2) - 1 = (4 \times 5) - 1 = 20 - 1 = 19$$

• Step 5: Write the rationalized expression.

  $$\frac{12+5\sqrt{5}}{19}$$

  The rationalized form is $\boxed{\frac{12+5\sqrt{5}}{19}}$.