Given:
Part A: [tex]\frac{\left(17^{3}\right)^{6} \cdot 17^{-10}}{17^{8}}[/tex]
Part B: [tex]\left(17^{6}\right)^{3} \cdot 17^{-9}[/tex]
To find:
The value of each expression.
Solution:
Part A:
Using exponent rule: [tex](a^m)^n=a^{mn}[/tex]
[tex]$\frac{\left(17^{3}\right)^{6} \cdot 17^{-10}}{17^{8}}=\frac{\left(17\right)^{3\times 6} \cdot 17^{-10}}{17^{8}}[/tex]
[tex]$=\frac{(17)^{18} \cdot 17^{-10}}{17^{8}}[/tex]
Using exponent rule: [tex]a^m \cdot a^{n}= a^{m+n}[/tex]
[tex]$=\frac{(17)^{18+(-10)}}{17^{8}}[/tex]
[tex]$=\frac{17^{8}}{17^{8}}[/tex]
Cancel the common factor, we get
= 1
[tex]$\frac{\left(17^{3}\right)^{6} \cdot 17^{-10}}{17^{8}}=1[/tex]
Part B:
[tex]\left(17^{6}\right)^{3} \cdot 17^{-9}[/tex]
Using exponent rule: [tex](a^m)^n=a^{mn}[/tex]
[tex]\left(17^{6}\right)^{3} \cdot 17^{-9}=\left(17\right)^{6\times 3} \cdot 17^{-9}[/tex]
[tex]=(17)^{18} \cdot 17^{-9}[/tex]
Using exponent rule: [tex]a^m \cdot a^{n}= a^{m+n}[/tex]
[tex]=(17)^{18+(-9)}[/tex]
[tex]=17^9[/tex]
[tex]\left(17^{6}\right)^{3} \cdot 17^{-9}=17^9[/tex]
