Q.2 Simplify and express each of the following in exponential form:
(i) $$\frac{(2^3 × 3^4 × 4)}{(3 × 32)}$$
(ii) $$[(5^2)^3 × 5^4]÷ 5^7$$
(iii) $$25^4÷5^3$$
(iv) $$\frac{3×7^2×11^8}{21×11^3}$$
(v) $$\frac{3^7}{3^4×3^3}$$
(vi) $$2^0+3^0+4^0$$
(vii) $$2^0×3^0×4^0$$
(viii) $$(3^0+2^0)×5^0$$
(ix)$$\frac{2^8×a^5}{4^3×a^3}$$
(x) $$(\frac{a^5}{a^3} ) × a^8$$
(xi) $$\frac{4^5×a^8b^3}{4^5×a^5b^2}$$
(xii) $$(2^3× 2)^2$$

(i) $$\frac{2^3 × 3^4 × 4}{3 × 32}$$ = $$\frac{2^3 × 3^4 × 2^2}{3 × 2^5}$$
= $$\frac{2^{3+2} × 3^4}{3 × 2^5}$$=$$\frac{2^{5} × 3^4}{3 × 2^5}$$
= $$2^{5-5}×3^{4-1}$$ = $$2^{0}×3^{3}$$= $$3^3$$
(ii)$$[(5^2)^3 × 5^4]÷ 5^7$$ = $$(5^6 × 5^4)÷ 5^7$$
= $$(5^{6+4} ÷ 5^7$$ = $$5^{10-7}$$ = $$5^3$$
(iii) $$25^4÷5^3$$ = $$(5^2)^4÷5^3$$= $$5^8÷5^3$$ = $$5^{8-3}$$= $$5^5$$
(iv) $$\frac{3×7^2×11^8}{21×11^3}$$ = $$\frac{3×7^2×11^8}{7×3×11^3}$$
= $$7^{2-1}×11^{8-3}$$ = $$7×11^{5}$$ \) (v) $$\frac{3^7}{3^4×3^3}$$ = $$\frac{3^7}{3^{4+3}}$$ = $$\frac{3^7}{3^7}$$ = 1
(vi) $$2^0+3^0+4^0$$ = 1+1+1 = 0$$[ \therefore a^0= 1 ]$$
(vii) $$2^0×3^0×4^0$$ = 1×1×1 = 1
(viii) $$(3^0+2^0)×5^0$$ = (1+1)×1=2×1=2
(ix) $$\frac{2^8×a^5}{4^3×a^3}$$ = $$\frac{2^8×a^5}{(2^2)^3×a^3}$$
= $$\frac{2^8×a^5}{2^6×a^3}$$ = $$2^{8-6}{×a^{5-3}$$ = $$2^2×a^2$$ = $$(2a)^2$$
(x) $$(\frac{a^5}{a^3} ) × a^8$$ = $$\frac{a^{5+8}}{a^3}$$ = $$\frac{a^{13}}{a^3}$$
= $$a^{13-3}$$ = $$a^{10}$$
(xi) $$\frac{4^5×a^8b^3}{4^5×a^5b^2}$$ = $$\frac{a^8b^3}{a^5b^2}$$
= $$a^{8-5}b^{3-2}$$ = $$a^3b$$
(xii) $$(2^3× 2)^2$$ = $$2^6 ×2^2$$ = $$2^{6+2}$$ = $$2^{8}$$