Final answer:
To simplify ((2a²)/(3a⁶b²))², square the numerator and the denominator terms, then combine the exponents by subtracting. The final simplified expression is (4)/(9a⁶b⁴), using only positive exponents.
Step-by-step explanation:
To simplify the expression ((2a²)/(3a⁶b²))² using only positive exponents, first apply the power of 2 to each term inside the parentheses. This means you have to square the numerator and the denominator separately. The rule for squaring exponentials states that you should square the digit term in the usual way and multiply the exponent of the exponential term by 2.
For the numerator (2a²)², the 2 is squared to get 4, and the a² is squared to get a⁴ (since 2 times 2 equals 4). For the denominator (3a⁶b²)², the 3 is squared to get 9, a⁶ is squared to get a¹ (since 6 times 2 equals 12), and b² is squared to get b⁴. This yields:
(4a⁴)/(9a¹b⁴)
Next, reduce the fraction by dividing the exponential terms with like bases by subtracting the smaller exponent from the larger exponent for each base. Since a¹ is in the denominator, you are left with a⁶ in the denominator after dividing a⁴ by a¹. The b⁴ remains in the denominator as there is no b term in the numerator to cancel it out. So the final simplified expression is:
(4)/(9a⁶b⁴).