1.2 Exponents and scientific notation  (Page 4/9)

1. $b^2\cdot b^{\mathrm{-8}}=b^{2-8}=b^{\mathrm{-6}}=\frac{1}{b^6}$
2. ${\left(-x\right)}^5\cdot {\left(-x\right)}^{\mathrm{-5}}={\left(-x\right)}^{5-5}={\left(-x\right)}^0=1$
3. $\frac{\mathrm{-7}z}{{\left(\mathrm{-7}z\right)}^5}=\frac{{\left(\mathrm{-7}z\right)}^1}{{\left(\mathrm{-7}z\right)}^5}={\left(\mathrm{-7}z\right)}^{1-5}={\left(\mathrm{-7}z\right)}^{\mathrm{-4}}=\frac{1}{{\left(\mathrm{-7}z\right)}^4}$

Use the product and quotient rules and the new definitions to simplify each expression.

1. ${\left(a{b}^2\right)}^3={\left(a\right)}^3\cdot {\left(b^2\right)}^3=a^{1\cdot 3}\cdot b^{2\cdot 3}=a^3{b}^6$
2. $2t^{15}={\left(2\right)}^{15}\cdot {\left(t\right)}^{15}=2^{15}{t}^{15}=32,768t^{15}$
3. ${\left(\mathrm{-2}{w}^3\right)}^3={\left(\mathrm{-2}\right)}^3\cdot {\left(w^3\right)}^3=\mathrm{-8}\cdot w^{3\cdot 3}=\mathrm{-8}{w}^9$
4. $\frac{1}{{\left(\mathrm{-7}z\right)}^4}=\frac{1}{{\left(\mathrm{-7}\right)}^4\cdot {\left(z\right)}^4}=\frac{1}{2,401z^4}$
5. ${\left(e^{\mathrm{-2}}{f}^2\right)}^7={\left(e^{\mathrm{-2}}\right)}^7\cdot {\left(f^2\right)}^7=e^{\mathrm{-2}\cdot 7}\cdot f^{2\cdot 7}=e^{\mathrm{-14}}{f}^{14}=\frac{f^{14}}{e^{14}}$

1. ${\left(\frac{4}{z^{11}}\right)}^3=\frac{{\left(4\right)}^3}{{\left(z^{11}\right)}^3}=\frac{64}{z^{11\cdot 3}}=\frac{64}{z^{33}}$
2. ${\left(\frac{p}{q^3}\right)}^6=\frac{{\left(p\right)}^6}{{\left(q^3\right)}^6}=\frac{p^{1\cdot 6}}{q^{3\cdot 6}}=\frac{p^6}{q^{18}}$
3. ${\left(\frac{\mathrm{-1}}{t^2}\right)}^{27}=\frac{{\left(\mathrm{-1}\right)}^{27}}{{\left(t^2\right)}^{27}}=\frac{\mathrm{-1}}{t^{2\cdot 27}}=\frac{\mathrm{-1}}{t^{54}}=-\frac{1}{t^{54}}$
4. ${\left(j^3{k}^{\mathrm{-2}}\right)}^4={\left(\frac{j^3}{k^2}\right)}^4=\frac{{\left(j^3\right)}^4}{{\left(k^2\right)}^4}=\frac{j^{3\cdot 4}}{k^{2\cdot 4}}=\frac{j^{12}}{k^8}$
5. ${\left(m^{\mathrm{-2}}{n}^{\mathrm{-2}}\right)}^3={\left(\frac{1}{m^2{n}^2}\right)}^3=\frac{{\left(1\right)}^3}{{\left(m^2{n}^2\right)}^3}=\frac{1}{{\left(m^2\right)}^3{\left(n^2\right)}^3}=\frac{1}{m^{2\cdot 3}\cdot n^{2\cdot 3}}=\frac{1}{m^6{n}^6}$