The week 3 slides define an unambiguous CFG for a small arithmetic language (where "n" is taken to be any natural number):

E --> P E --> E + P P --> A P --> P * A A --> (E) A --> n

In this grammar, both the + and the * operator are left-associative, and * has a higher precedence than +. This is enforced by the rules of the grammar, which divides arithmetic expressions into three categories:

- E represents expressions involving the + operator, only allowing other E expressions on the right if they're within parentheses.

- P represents expressions involving the * operator, only allowing other P expressions on the right or E expressions on either side if they're within parentheses.

- A represents numeric literals or parenthesized expressions.

Here are some example leftmost derivations using this CFG:

"E" -> "P" -> "A" -> "10"

"E" -> "E + P" -> "P + P" -> "A + P" -> "1 + P" -> "1 + A" -> "1 + 2"

"E" -> "E + P" -> "E + P + P" -> "P + P + P" -> "A + P + P" -> "1 + P + P" -> "1 + A + P" -> "1 + (E) + P" -> "1 + (P) + P" -> "1 + (P * A) + P" -> "1 + (A * A) + P" -> "1 + (2 * A) + P" -> "1 + (2 * 3) + P" -> "1 + (2 * 3) + A" -> "1 + (2 * 3) + 4"

Modify the CFG to add a new nonterminal X, representing expressions involving an infix exponent operator (^), so that the ^ operator is right-associative and has higher precedence than both + and *. The CFG should remain unambiguous, so that there is exactly one valid leftmost derivation that produces any given expression in the language. Give two example leftmost derivations that involve the ^ operator.

Please as clear be as possible.