Final answer:
To simplify (27x¹y¹⁵)¹/³, take the cube root of 27 to get 3, then apply the fractional exponent 1/3 to the variables which results in multiplying their exponents by 1/3. The simplified form is 3x³y⁵.
Step-by-step explanation:
To simplify the expression (27x¹y¹⁵)¹/³, we need to apply the properties of rational exponents. According to these properties, when we raise an expression inside parentheses to a fractional power, the power extends to both the numerical coefficient and the variable parts.
Let's start by simplifying the numerical coefficient:
- 27 to the power of 1/3 is essentially the cube root of 27, which is 3 because 3³ = 27.
Now let's apply the exponent 1/3 to the variables:
- The exponent on x is 9, so x¹ to the power of 1/3 can be calculated by multiplying the exponents (9 * 1/3), giving us x³.
- Similarly, the exponent on y is 15, so y¹⁵ to the power of 1/3 also involves multiplying the exponents (15 * 1/3), which results in y⁵.
Combining these results, we get the simplified expression: 3x³y⁵.