1. **State the problem:** (a) Given that $$4^3 \times 16^{\frac{1}{4}} = 2^x$$, find the value of $$x$$. 2. **Recall the formula and rules:** - Express all terms with the same base to simplify. - Use the laws of exponents: $$a^m \times a^n = a^{m+n}$$ and $$\left(a^m\right)^n = a^{mn}$$. 3. **Rewrite the bases:** - $$4 = 2^2$$, so $$4^3 = (2^2)^3 = 2^{2 \times 3} = 2^6$$. - $$16 = 2^4$$, so $$16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \times \frac{1}{4}} = 2^1 = 2$$. 4. **Multiply the terms:** $$4^3 \times 16^{\frac{1}{4}} = 2^6 \times 2^1 = 2^{6+1} = 2^7$$. 5. **Equate powers:** Since $$4^3 \times 16^{\frac{1}{4}} = 2^x$$ and we found it equals $$2^7$$, then $$x = 7$$. --- 6. **State the problem:** (b)(i) Simplify $$3p^3 \times (4p^4)^3$$. 7. **Recall the rules:** - Use $$\left(a^m\right)^n = a^{mn}$$. - Multiply coefficients and add exponents of like bases. 8. **Simplify inside the parentheses:** $$(4p^4)^3 = 4^3 \times (p^4)^3 = 64p^{12}$$. 9. **Multiply:** $$3p^3 \times 64p^{12} = (3 \times 64) p^{3+12} = 192p^{15}$$. --- 10. **State the problem:** (b)(ii) Simplify $$\frac{3q^2}{2} \div \frac{15q}{8}$$. 11. **Recall the rule:** Dividing by a fraction is multiplying by its reciprocal. 12. **Rewrite division as multiplication:** $$\frac{3q^2}{2} \times \frac{8}{15q}$$. 13. **Multiply numerators and denominators:** $$\frac{3q^2 \times 8}{2 \times 15q} = \frac{24q^2}{30q}$$. 14. **Simplify the fraction:** $$\frac{24q^2}{30q} = \frac{\cancel{6} \times 4 q^{\cancel{2}}}{\cancel{6} \times 5 q^{\cancel{1}}} = \frac{4q}{5}$$. --- 15. **State the problem:** (b)(iii) Simplify $$\sqrt{\frac{r^6}{r^4}}$$. 16. **Simplify inside the square root:** $$\frac{r^6}{r^4} = r^{6-4} = r^2$$. 17. **Take the square root:** $$\sqrt{r^2} = |r|$$ (absolute value of $$r$$). --- **Final answers:** (a) $$x = 7$$ (b)(i) $$192p^{15}$$ (b)(ii) $$\frac{4q}{5}$$ (b)(iii) $$|r|$$