Final answer:
To simplify ((t²)/(t⁻⁶))¹⁄₂, we subtract the denominator's exponent from the numerator's, giving us t² + 6 = t⁸. Then we raise t⁸ to the 1/2 power, which means multiplying the exponent by 1/2, simplifying to t⁴.
Step-by-step explanation:
To simplify the expression ((t²)/(t⁻⁶))¹⁄₂ by using positive exponents only, we can begin by tackling the division of exponentials. We take the original exponents and subtract the exponent in the denominator from the exponent in the numerator, as per the rules for division of exponentials.
In this case, we have t² in the numerator and t⁻⁶ in the denominator. Using the rule for division that tells us to subtract the bottom exponent from the top exponent, we get t² - (-6) = t² + 6 = t⁸. Now, we have (t⁸)¹⁄₂, which means we need to raise t⁸ to the 1/2 power.
Cubing of exponentials suggests we multiply the exponent by the fractional power. Here, we're dealing with a square root, which is the same thing as raising to the power of 1/2. So, we multiply 8 by 1/2 to get t⁴.
The simplified expression with positive exponents only is t⁴.