Answer:
- 1/32
- 32
- -(1/32)
- -(1/32)
Step-by-step explanation:
1.
[tex]2^-^1 \cdot 2^-^4\\\\\mathrm{Apply\:exponent\:rule}:\quad \:a^b\cdot \:a^c=a^{b+c}\\2^{-1}\cdot \:2^{-4}=2^{-1-4}\\\\=2^{-1-4}\\\\=2^{-5}\\\\\mathrm{Apply\:exponent\:rule}:\quad \:a^{-b}=\frac{1}{a^b}\\2^{-5}=\frac{1}{2^5}\\\\2^5=32\\\\=\frac{1}{32}[/tex]
2.
[tex]\frac{2}{2^-^4} \\\\\mathrm{Cancel\:the\:common\:factor:}\:2^{-4}\\=2^5\\\\2^5=32\\\\=32[/tex]
3.
[tex](-\frac{1}{2} )^3\cdot (-\frac{1}{2} )^2\\\\\mathrm{Apply\:exponent\:rule}:\quad \:a^b\cdot \:a^c=a^{b+c}\\\left(-\frac{1}{2}\right)^3\left(-\frac{1}{2}\right)^2=\left(-\frac{1}{2}\right)^{3+2}\\\\=\left(-\frac{1}{2}\right)^5\\\\\mathrm{Apply\:exponent\:rule}:\quad \left(-a\right)^n=-a^n,\:\quad \mathrm{if\:}n\mathrm{\:is\:odd}\\\\\left(-\frac{1}{2}\right)^5=-\left(\frac{1}{2}\right)^5\\\\\mathrm{Apply\:exponent\:rule}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}\\[/tex]
[tex]\left(\frac{1}{2}\right)^5=\frac{1^5}{2^5}\\\\=-\frac{1^5}{2^5}\\\\=-\frac{1}{32}[/tex]
4.
[tex]\frac{(-2)^-^5}{(-2)^-^1^0} \\\\\mathrm{Apply\:rule}\:a^0=1,\:a\ne \:0\\\left(\left(-2\right)^{-1}\right)^0=1\\\\=\frac{\left(-2\right)^{-5}}{1}\\\\\left(-2\right)^{-5}=-2^{-5}\\\\=\frac{-2^{-5}}{1}\\\\\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{-a}{b}=-\frac{a}{b}\\\\=-\frac{2^{-5}}{1}\\\\\mathrm{Apply\:rule}\:\frac{a}{1}=a\\=-2^{-5}\\\\\mathrm{Simplify}\:2^{-5}:\quad \frac{1}{32}\\=-\frac{1}{32}[/tex]
