### Instruction: Express the value of the following expression as a fraction. $$1+\cfrac{2}{3+\cfrac{4}{5}}$$.### Response: I notice that this expression has a fraction within a fraction, which is called a nested fraction.
To simplify it, I can start by rewriting the innermost fraction as a single term with a common denominator.
That is, $$3+\cfrac{4}{5}=\cfrac{15}{5}+\cfrac{4}{5}=\cfrac{19}{5}$$
Now I can replace the denominator of the outer fraction with this value, and get $$1+\cfrac{2}{\cfrac{19}{5}}$$
To simplify this further, I can use the rule that dividing by a fraction is the same as multiplying by its reciprocal.
That means I can rewrite the expression as $$1+\cfrac{2}{\cfrac{19}{5}}=1+\cfrac{2}{1}\cdot\cfrac{5}{19}$$
Now I can multiply the numerators and the denominators of the fractions, and get $$1+\cfrac{2}{1}\cdot\cfrac{5}{19}=1+\cfrac{10}{19}$$
Finally, to express this as a single fraction, I can find a common denominator for 1 and $\cfrac{10}{19}$.
That is, $$1+\cfrac{10}{19}=\cfrac{19}{19}+\cfrac{10}{19}=\cfrac{29}{19}$$