(a) Simplify.
$\frac{x^{\frac{2}{3}}}{x^{\frac{8}{3}}}$
(b) $16 = 64^k$
Find the value of $k$.
(c) Solve.
$3^{3x} \times \left( \frac{1}{9} \right)^{4-3x} = 3$
▶️ Answer/Explanation
(a) Ans: $\frac{1}{x^2}$ or $x^{-2}$
Using laws of indices: $x^{\frac{2}{3} – \frac{8}{3}} = x^{-2} = \frac{1}{x^2}$
(b) Ans: $\frac{2}{3}$
Express as powers of 4: $4^2 = 4^{3k}$ → $2 = 3k$ → $k = \frac{2}{3}$
(c) Ans: 1
Rewrite $\frac{1}{9}$ as $3^{-2}$: $3^{3x} \times 3^{-2(4-3x)} = 3^1$
Combine exponents: $3x – 8 + 6x = 1$ → $9x = 9$ → $x = 1$
(a) $\frac{10x^2 – 60x}{x^2 – x – 30}$
(b) $\frac{7}{x + 3} + \frac{5}{8x – 1}$
▶️ Answer/Explanation
(a) $\frac{10x}{x + 5}$
(b) $\frac{61x + 8}{(x + 3)(8x – 1)}$
(a) Factor numerator: $10x(x – 6)$, denominator: $(x – 6)(x + 5)$
Cancel $(x – 6)$ to get $\frac{10x}{x + 5}$
(b) Common denominator: $(x + 3)(8x – 1)$
Combine: $\frac{7(8x – 1) + 5(x + 3)}{(x + 3)(8x – 1)} = \frac{61x + 8}{(x + 3)(8x – 1)}$