### Instruction: Express the value of the following expression as a fraction. $$1+\cfrac{2}{3+\cfrac{4}{5}}$$.### Response: I notice that this expression is a type of continued fraction, where a fraction appears in the denominator of another fraction, and so on.
A common way to simplify continued fractions is to work from the inside out, starting with the innermost fraction and using the identity $$a+\cfrac{b}{c}=\cfrac{ac+b}{c}$$
So, I apply this identity to the innermost fraction, $\cfrac{4}{5}$, and the term that precedes it, 3, and get $$3+\cfrac{4}{5}=\cfrac{3\times 5+4}{5}=\cfrac{19}{5}$$
Now, I can replace the innermost fraction with this equivalent expression, and get $$1+\cfrac{2}{\cfrac{19}{5}}$$
To simplify this further, I can use another identity, $$\cfrac{a}{\cfrac{b}{c}}=\cfrac{ac}{b}$$
So, I apply this identity to the fraction in the denominator, $\cfrac{19}{5}$, and the term that precedes it, 2, and get $$\cfrac{2}{\cfrac{19}{5}}=\cfrac{2\times 5}{19}=\cfrac{10}{19}$$
Now, I can replace the fraction in the denominator with this equivalent expression, and get $$1+\cfrac{10}{19}$$