A rational number is defined as the quotient of two integers `a`

and `b`

, called the numerator and denominator, respectively, where `b != 0`

.

The absolute value `|r|`

of the rational number `r = a/b`

is equal to `|a|/|b|`

.

The sum of two rational numbers `r₁ = a₁/b₁`

and `r₂ = a₂/b₂`

is `r₁ + r₂ = a₁/b₁ + a₂/b₂ = (a₁ * b₂ + a₂ * b₁) / (b₁ * b₂)`

.

The difference of two rational numbers `r₁ = a₁/b₁`

and `r₂ = a₂/b₂`

is `r₁ - r₂ = a₁/b₁ - a₂/b₂ = (a₁ * b₂ - a₂ * b₁) / (b₁ * b₂)`

.

The product (multiplication) of two rational numbers `r₁ = a₁/b₁`

and `r₂ = a₂/b₂`

is `r₁ * r₂ = (a₁ * a₂) / (b₁ * b₂)`

.

Dividing a rational number `r₁ = a₁/b₁`

by another `r₂ = a₂/b₂`

is `r₁ / r₂ = (a₁ * b₂) / (a₂ * b₁)`

if `a₂`

is not zero.

Exponentiation of a rational number `r = a/b`

to a non-negative integer power `n`

is `r^n = (a^n)/(b^n)`

.

Exponentiation of a rational number `r = a/b`

to a negative integer power `n`

is `r^n = (b^m)/(a^m)`

, where `m = |n|`

.

Exponentiation of a rational number `r = a/b`

to a real (floating-point) number `x`

is the quotient `(a^x)/(b^x)`

, which is a real number.

Exponentiation of a real number `x`

to a rational number `r = a/b`

is `x^(a/b) = root(x^a, b)`

, where `root(p, q)`

is the `q`

th root of `p`

.

Implement the following operations:

- addition, subtraction, multiplication and division of two rational numbers,
- absolute value, exponentiation of a given rational number to an integer power, exponentiation of a given rational number to a real (floating-point) power, exponentiation of a real number to a rational number.

Your implementation of rational numbers should always be reduced to lowest terms. For example, `4/4`

should reduce to `1/1`

, `30/60`

should reduce to `1/2`

, `12/8`

should reduce to `3/2`

, etc. To reduce a rational number `r = a/b`

, divide `a`

and `b`

by the greatest common divisor (gcd) of `a`

and `b`

. So, for example, `gcd(12, 8) = 4`

, so `r = 12/8`

can be reduced to `(12/4)/(8/4) = 3/2`

.

Assume that the programming language you are using does not have an implementation of rational numbers.

Last updated 22 September 2021

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