Question

Simplify: ${\left[\frac{2^2}{15}\right]}^4\times {\left[\frac{5^3}{2^3}\right]}^3+{\left(\frac{5}{9}\right)}^3$

Answer 1

${\left[\frac{2^2}{15}\right]}^4\times {\left[\frac{5^3}{2^3}\right]}^3+{\left(\frac{5}{9}\right)}^3$

we have the property of exponents

$\Rightarrow {\left(a^m\right)}^n=a^{mn}$

and also we have the BODMAS rule to solve complex algebraic expression. ( first solve the bracket , followed by division, multiplication, addition and then subtraction).

Using the above laws we can simplify as,

${\left[\frac{2^2}{15}\right]}^4\times {\left[\frac{5^3}{2^3}\right]}^3+{\left(\frac{5}{9}\right)}^3$

$=\left(\frac{2^{2\times 4}}{{15}^4}\right)\times \left(\frac{5^{3\times 3}}{2^{3\times 3}}\right)+\frac{5^3}{9^3}$

$=\frac{2^8}{{15}^4}\times \frac{5^9}{2^9}+\frac{5^3}{9^3}$

$=\frac{5\times 5^8}{15\times {15}^3\times 2}+\frac{5^3}{9^3}$

$=\frac{568}{6\times {15}^3}+\frac{5^3}{9^3}$

$=\frac{5^8}{6\times {(3\times 5)}^3}+\frac{5^3}{9^3}$

$=\frac{5^8}{6\times 3^3\times 5^3}+\frac{5^3}{9^3}$

$=\frac{5^5}{6\times 3^3}+\frac{5^3}{3^6}$ $(\therefore 9=3^2\Rightarrow 9^3=3^{2\times 3}=3^6)$

$=\frac{5^3}{3^3}[\frac{5^2}{6}+\frac{1}{3^3}]$

$=\frac{5^3}{3^3}[\frac{25}{6}+\frac{1}{27}]$

$=\frac{5^3\times 681}{3^3\times 6\times 27}$

$=\frac{5^3\times 227}{3^3\times 2\times 27}$

$\cong \mathrm{19.46.}$

Hence, the answer is $\mathrm{19.46.}$