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Fractions represent a part of a whole. When working with fractions, we often need to simplify them to make our calculations easier. To simplify fractions, you divide the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD).

In the exercise, the first fraction \(\frac{32 y^{5}}{8 y^{2}}\) was simplified by dividing both the numerator and the denominator by 8. Similarly, the second fraction \(\frac{60 y^{10}}{5 y^{7}}\) was simplified by dividing both numbers by 5. Simplifying fractions helps in reducing complex fractions into more manageable numbers.

Remember to always look for common factors between the numerator and the denominator to simplify your work. This makes subtraction or addition of fractions more straightforward.

Exponents are a way to express repeated multiplication of the same number. The expression \(y^{5}\) means \(y \times y \times y \times y \times y\). When simplifying fractions with exponents, you can subtract the exponents if the bases (the number or variable being multiplied) are the same, which is a rule from exponent properties.

For example, in the exercise, the fraction \(\frac{32 y^{5}}{8 y^{2}}\) was simplified to \(\frac{32}{8} \times \frac{y^{5}}{y^{2}} = 4y^{3}\). Here, we divided the coefficients (numbers) and subtracted the exponents since \(y^{5}/y^{2} = y^{5-2} = y^{3}\).

Similarly, for \(\frac{60 y^{10}}{5 y^{7}}\), it became \(\frac{60}{5} \times \frac{y^{10}}{y^{7}} = 12y^{3}\). Understanding exponent rules is crucial in manipulating terms and simplifying algebraic expressions efficiently.

Like terms in algebra are terms that have the same variable raised to the same power. Only like terms can be combined by adding or subtracting their coefficients (the numerical part of the term) while keeping the variable part unchanged.

In the exercise, after simplifying each fraction: \(4y^{3}\) and \(12y^{3}\) were combined. Since they are both \(y^{3}\) terms, you subtract their coefficients: \(4 - 12 = -8\), giving us \(-8y^{3}\).

Always ensure that you group and combine only the like terms in any algebraic expression. This keeps your work accurate and helps in further simplification or solving of equations.