Answer: The answer is: " [tex]\frac{1}{80}[/tex] " .
________
Step-by-step explanation:
________
Given: " [tex]{a^{3} = 5[/tex] " ; and: " [tex]b^{3} = 4[/tex] " ;
Find: " [tex](ab^{2})^{-3}[/tex] " ;
________
Note:
" [tex](ab^2)^-^3 = a^-^3 b^(^2^*^-^3^) = a^-^3b^-^6 = \frac{1}{a^3} * \frac{1}{b^6}[/tex] ;
________
Take note of the following properties of exponents:
_______________________
{Note: " [tex](xy)^z = x^zy^z[/tex] ".
{Note: " [tex](x^y)^z = x^(^y^*^z^)[/tex] ".
{Note: " [tex]x^(^-^y^) = \frac{1}{x^y}[/tex] " ; Assuming " [tex]x\neq 0[/tex] " .}.
______________________
We have: " [tex]\frac{1}{a^3}*\frac{1}{b^6}[/tex] " ;
________
Given: " [tex]a^3 = 5[/tex] " ; We can substitute:
________
" [tex]\frac{1}{a^3} = \frac{1}{5}[/tex] " ;
________
Now, regarding: " [tex]b^6[/tex] " ;
________
Given: " [tex]b^{3} = 4[/tex] " ;
Note that " 6 ÷ 3 = 2 " ;
________
So: " [tex]b^{6} = (b^3)^2[/tex] " ; {since: " [tex](a^m)^n = a^(^m^*^n^)[/tex] " } ;
________
So: We can substitute "4" for " [tex]b^{3}[/tex] " ;
and solve for the value of: " [tex]b^{6}[/tex] " :
________
" [tex]b^6 = (b^3)^2 = 4^2 = (4*4) = 16[/tex] " ;
So: " [tex]\frac{1}{b^6} = \frac{1}{16}[/tex] " ;
________
Now, " [tex]\frac{1}{a^3}*\frac{1}{b^6}[/tex] = [tex]\frac{1}{5} * \frac{1}{16} = \frac{(1*1)}{(5*16)} = \frac{1}{80}[/tex] " .
________
The answer is: " [tex]\frac{1}{80}[/tex] " .
________
Hope this is helpful to you!
Wishing you the best!
________
