Question

Simplify and express each of the following in exponential form:
(i) $\left\{\right(3^2{)}^3\times 2^6\}\times 5^6$
(ii) ${\left(\frac{x}{y}\right)}^{12}\times y^{24}\times (2^3{)}^4$
(iii) ${\left(\frac{5}{2}\right)}^6\times {\left(\frac{5}{2}\right)}^2$
(iv) ${\left(\frac{2}{3}\right)}^5\times {\left(\frac{3}{5}\right)}^5$

Answer 1

We have
(i) {(3²)³ x 2⁶} x 5⁶
= {3⁶ x 2⁶} x 5⁶ [since (a^m)ⁿ = a^mn]
= 6⁶ x 5⁶ [since a^m x b^m = (a x b)^m ]
= 30⁶

(ii)
${\left(\frac{x}{y}\right)}^{12}\times y^{24}\times (2^3{)}^4$
$=\frac{x^{12}}{y^{12}}\times y^{24}\times 2^{12}$
$=x^{12}\times \frac{y^{24}}{y^{12}}\times 2^{12}$
$=x^{12}\times y^{24-12}\times 2^{12}$
$=x^{12}\times y^{12}\times 2^{12}$
$=(2xy{)}^{12}[\mathrm{since}{a}^m\times b^m\times c^m=(a\times b\times c{)}^m]$
(iii)
${\left(\frac{5}{2}\right)}^6\times {\left(\frac{5}{2}\right)}^2$
$={\left(\frac{5}{2}\right)}^8[\mathrm{since}{a}^m\times a^n=a^{m+n}]$

(iv)
${\left(\frac{2}{3}\right)}^5\times {\left(\frac{3}{5}\right)}^5$
$={\left(\frac{2}{3}\times \frac{3}{5}\right)}^5[\mathrm{since}{a}^m\times b^m=(a\times b{)}^m]$
$={\left(\frac{2}{5}\right)}^5$