Question

Simplifyand express each of the following in exponential form:

(i) $\frac{2^3\times 3^4\times 4}{3\times 32}$

(ii) $({5^2}^3\times 5^4)÷5^7$

(iii) ${25}^4÷5^3$

(iv) $\frac{3\times 7^2\times {11}^8}{21\times {11}^3}$

(v) $\frac{3^7}{3^4\times 3^3}$

(vi) $2^0+3^0+4^0$

(vii) $2^0\times 3^0\times 4^0$

(viii) $(3^0+2^0)\times 5^0$

(ix) $\frac{2^8\times a^5}{4^3\times a^3}$

(x) $\frac{a^5}{a^3}\times a^8$

(xi) $\frac{4^5\times a^8{b}^3}{4^5\times a^5{b}^2}$

(xii) $(2^3\times 2{)}^2$

Answer 1

(i) $\frac{2^3\times 3^4\times 4}{3\times 32}$

$=\frac{2^3\times 3^4\times 2^2}{3\times 2^5}$

$=\frac{2^{3+2}\times 3^4}{3^1\times 2^5}$, [$\because a^m\times a^n=a^{m+n},a\ne 0$]

$=\frac{2^5\times 3^{4-1}}{2^5}$ [$\because a^m÷a^n=a^{m-n},a\ne 0$]

$=3^3$

$\therefore \frac{2^3\times 3^4\times 4}{3\times 32}=3^3$

(ii) $({5^2}^3\times 5^4)÷5^7$

$=(5^8\times 5^4)÷5^7$

$=\left(5^{8+4}\right)÷5^7$

$=5^{12}÷5^7$

$=5^{12-7}$

$=5^5$

$\therefore ({5^2}^3\times 5^4)÷5^7=5^5$

(iii) ${25}^4÷5^3$

$=(5^2{)}^4÷5^3$

$=5^8÷5^3$ [ $\because a^m{)}^n=a^{mn},a\ne 0$]

$=5^{8-3}$

$=5^5$

$\therefore {25}^4÷5^3=5^5$

(iv) $\frac{3\times 7^2\times {11}^8}{21\times {11}^3}$

$=\frac{7^2\times {11}^8}{7\times {11}^3}$

$=7^{2-1}\times {11}^{8-3}$

$=7\times {11}^5$

$\therefore \frac{3\times 7^2\times {11}^8}{21\times {11}^3}=7\times {11}^5$

(v) $\frac{3^7}{3^4\times 3^3}$

$=\frac{3^7}{3^{4+3}}$

$=\frac{3^7}{3^7}$

$=1$

$\therefore \frac{3^7}{3^4\times 3^3}=1$

(vi) $2^0+3^0+4^0$

$=1+1+1$, [$\because a^0=1,a\ne 0$]

$=3$

$\therefore 2^0+3^0+4^0=3$

vii) $2^0\times 3^0\times 4^0$

$=1\times 1\times 1$

$=1$

$\therefore 2^0\times 3^0\times 4^0=1$

(viii) $(3^0+2^0)\times 5^0$

$=(1+1)\times 1$

$=2\times 1$

$=2$

$\therefore (3^0+2^0)\times 5^0=2$

(ix) $\frac{2^8\times a^5}{4^3\times a^3}$

$=\frac{2^8\times a^5}{(2^2{)}^3\times a^3}$

$=\frac{2^8\times a^5}{2^6\times a^3}$

$=2^{8-6}\times a^{5-3}$

$=2^2\times a^2$

$=(2a{)}^2$, [$\because a^m\times b^m=(ab{)}^m$]

$\therefore \frac{2^8\times a^5}{4^3\times a^3}=(2a{)}^2$

(x) $\frac{a^5}{a^3}\times a^8$

$=a^{5-3}\times a^8$

$=a^2\times a^8$

$=a^{2+8}$

$=a^{10}$

$\therefore \frac{a^5}{a^3}\times a^8=a^{10}$

(xi) $\frac{4^5\times a^8{b}^3}{4^5\times a^5{b}^2}$

$=\frac{a^8{b}^3}{a^5{b}^2}$

$=a^{8-5}{b}^{3-2}$

$=a^{3}b$

$\therefore \frac{4^5\times a^8{b}^3}{4^5\times a^5{b}^2}=a^{3}b$

(xii) $(2^3\times 2{)}^2$

$=(2^{3+1}{)}^2$

$=(2^4{)}^2$

$=2^8$

$\therefore (2^3\times 2{)}^2=2^8$