Simplify the expression. Write your answer with positive exponents only. (4a^3b)^-2/(a^2b^6)^-3
To simplify this expression, we can use the rule of negative exponents, which states that any term raised to a negative exponent becomes the reciprocal of that term raised to the positive exponent.
Using this rule, we can rewrite the expression as:
((4a^3b)^-2)/((a^2b^6)^-3)
= ((1/(4a^3b)^2))/(1/(a^2b^6)^3)
= ((1/(16a^6b^2))/(1/(a^6b^18))
Now, to divide two fractions, we can multiply the first fraction by the reciprocal of the second fraction:
= (1/(16a^6b^2)) * ((a^6b^18)/1)
= (1 * a^6b^18)/(16a^6b^2 * 1)
= a^6b^18 / 16a^6b^2
Now, we can simplify further by subtracting the exponents with the same bases:
= (a^6/a^6) * (b^18/b^2) / 16
Since a^6/a^6 equals 1, the expression becomes:
= b^18/b^2 /16
Finally, we can subtract the exponents of b:
= b^(18-2) / 16
Simplifying the exponents, we have:
= b^16 / 16
Therefore, the simplified expression is b^16 / 16.