Question

Simplify and express the result in power notation with positive exponent.
(i) $(-4{)}^5÷(-4{)}^8$

(ii) ${\left(\frac{1}{2^3}\right)}^2$

(iii) $(-3{)}^4\times {\left(\frac{5}{3}\right)}^4$

(iv) $(3^{-7}÷3^{-10})\times 3^{-5}$

(v) $2^{-3}\times (-7{)}^{(-3)}$

Answer 1

(i) $(-4{)}^5÷(-4{)}^8=\frac{(-4{)}^5}{(-4{)}^8}=(-4{)}^{5-8}=(-4{)}^{-3}=\frac{1}{(-4{)}^3}$

(ii) ${\left(\frac{1}{2^3}\right)}^2=\frac{1}{2^{3\times 2}}=\frac{1}{2^6}$

(iii) $(-3{)}^4\times {\left(\frac{5}{3}\right)}^4=(-1{)}^4\times (3{)}^4\times {\left(\frac{5}{3}\right)}^4=(-1{)}^4\times 5^4=5^4$

(iv) $(3^{-7}÷3^{-10})\times 3^{-5}=\left(\frac{3^{-7}}{3^{-10}}\right)\times (3{)}^{-5}=3^{-7+10}\times 3^{-5}$
$=3^{3-5}$
$=3^{-2}=\frac{1}{3^2}$

(v) $2^{-3}\times (-7{)}^{(-3)}=\frac{1}{2^3}\times \frac{1}{(-7{)}^3}={\left(\frac{1}{2\times (-7)}\right)}^3$
$=\frac{1}{(-14{)}^3}$