Q: Simplify (4x^6+8y^4) / (16x).

A:  Many different methods exist!

Method 1

This is the same as:

4x^6 / (16x) + 8y^4 / (16x)

Reduce the numbers and the letters where possible:

(1)  4x^6 / (16x) [reduce the 4 and the 16 and take one x out of the top an bottom]

= x^5 / 4 or 1/4 x^5

(2)  8y^4 / (16x) [only the 8 and 16 can reduce]

= y^4 / (2x)

So, (4x^6+8y^4) / (16x) = x^5 / 4 + y^4 / (2x)

Method 2

(4x^6+8y^4) / (16x)

Factor the top to reduce like terms in the end:

4(x^6 + 2y^4) / (16x)

Reduce the fraction:

(x^6 + 2y^4) / (4x),

Method 3:  Using negative exponents

(4x^6 + 8y^4) / (16x) =

4x^6 / (16x) + 8y^4 / (16x) [Reduce the numbers.  Then, subtract the exponent of the x in the denominator from the exponent of the x in the numerator.  If there is no x in the numerator, the exponent will be negative]

(1/4)x^(6-1) + (1/2)y^4*x^-1 =
(1/4)x^5 + (1/2)y^4*x^-1.