1. Problem: Simplify the expressions using exponent laws and compute numerical values where possible. 2. Recall the exponent multiplication rule: $$a^m \times a^n = a^{m+n}$$ and the power of a power rule: $$ (a^m)^n = a^{mn} $$. 3. a) Simplify $$(x^3)(x^{7/3})$$: $$x^{3 + \frac{7}{3}} = x^{\frac{9}{3} + \frac{7}{3}} = x^{\frac{16}{3}}$$ 4. b) The expression is incomplete, so we cannot solve it. 5. d) Simplify $$(k^{4.8})(k^3)$$: $$k^{4.8 + 3} = k^{7.8}$$ 6. e) Simplify $$16^{0.25}$$: Recall $$16 = 2^4$$, so: $$16^{0.25} = (2^4)^{0.25} = 2^{4 \times 0.25} = 2^1 = 2$$ 7. Problem 2: Simplify expressions with positive exponents. 8. a) Simplify $$(x^3)(x^{-2/3})$$: $$x^{3 + (-\frac{2}{3})} = x^{\frac{9}{3} - \frac{2}{3}} = x^{\frac{7}{3}}$$ 9. b) Simplify $$(81^{-0.5})$$: Recall $$81 = 3^4$$, so: $$81^{-0.5} = (3^4)^{-0.5} = 3^{4 \times (-0.5)} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$ 10. Problem 3: Evaluate without a calculator. 11. a) Evaluate $$8^{2/3}$$: Recall $$8 = 2^3$$, so: $$8^{2/3} = (2^3)^{2/3} = 2^{3 \times \frac{2}{3}} = 2^2 = 4$$ 12. b) Evaluate $$16^{1/4}$$: Recall $$16 = 2^4$$, so: $$16^{1/4} = (2^4)^{1/4} = 2^{4 \times \frac{1}{4}} = 2^1 = 2$$ 13. d) Evaluate $$(3^{1/6})(3^{5/6})$$: $$3^{\frac{1}{6} + \frac{5}{6}} = 3^1 = 3$$ 14. e) The expression is incomplete, so we cannot solve it. Final answers: 1a) $$x^{\frac{16}{3}}$$ 1d) $$k^{7.8}$$ 1e) $$2$$ 2a) $$x^{\frac{7}{3}}$$ 2b) $$\frac{1}{9}$$ 3a) $$4$$ 3b) $$2$$ 3d) $$3$$