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) \((2^3× 2)^2 \) = \(2^6 ×2^2 \) = \( 2^{6+2} \) = \(2^{8} \)