Simplifying powers

Statement

Simplify the following expressions: (a) $\dfrac{a^9}{a^5}$; (b) $\dfrac{20b^7}{15b^4}$; (c) $\dfrac{(-8c^5)(2c^2)}{(4c^3)(2c^2)}$.

Solution

(a) For a quotient of powers with the same base, subtract the exponents: $\frac{a^9}{a^5} = a^{9-5} = a^4.$

(b) Simplify the coefficients and the powers: $\frac{20b^7}{15b^4} = \frac{20}{15}\,b^{7-4} = \frac{4}{3}\,b^3.$

(c) Compute numerator and denominator separately: $\text{num} = (-8c^5)(2c^2) = -16c^7, \qquad \text{den} = (4c^3)(2c^2) = 8c^5,$ hence $\frac{-16c^7}{8c^5} = -2c^{7-5} = -2c^2.$

$\boxed{a^4 \ ;\quad \tfrac{4}{3}b^3 \ ;\quad -2c^2}$