Use the properties of exponents to simplify as much as possible
(8a^3b^6)^(1/3)/(16a^8b^-4)^(1/4)
(8a^3b^6)^(1/3) = 2ab^2
(16a^8b^-4)^(1/4) = 2a^2b^-1
2ab^2 / 2a^2b^-1 = a^-1b^3 = b^3/a
To simplify the expression, we can use the properties of exponents:
First, let's simplify the numerator, (8a^3b^6)^(1/3):
- When we have an exponent raised to another exponent, we multiply the exponents. In this case, (a^3)^(1/3) equals a^(3*(1/3)), which simplifies to a^1 or simply a.
- Similarly, (b^6)^(1/3) gives us b^(6*(1/3)), which simplifies to b^2.
- Now, we can substitute these simplified expressions back into the numerator to get 8ab^2.
Next, let's simplify the denominator, (16a^8b^-4)^(1/4):
- Similarly to what we did before, (a^8)^(1/4) simplifies to a^(8*(1/4)), which equals a^2.
- With a negative exponent, b^-4 can be rewritten as 1/b^4.
- Combining these terms, we get 16a^2/b^4.
Now, the simplified expression is (8ab^2) / (16a^2/b^4).
To divide fractions, we can multiply by the reciprocal of the denominator, so we have (8ab^2) * (b^4 / 16a^2).
Multiplying the numerators and the denominators, we get 8 (a * b^2 * b^4) / (16 * a^2).
Simplifying further, we have 8ab^6 / 16a^2.
Finally, we can simplify this fraction by canceling out common factors:
- 8 can be divided by 8.
- a can be canceled out from both the numerator and denominator.
- b^6 can be divided by b^2.
Thus, the simplified expression becomes b^4 / 2a.
Therefore, the expression (8a^3b^6)^(1/3) / (16a^8b^-4)^(1/4) simplifies to b^4 / 2a.