Final answer:
To simplify the 'cube root of x⁴', rewrite it as x⁴ raised to the power of 1/3, which is x⁴/3 or x¹³, corresponding to option c) x⁴¹³.
Step-by-step explanation:
The question asks us to simplify the radical expression 'cube root of x⁴'. To address this, we'll be using properties of exponents that relate radicals to fractional powers.
The cube root of a number can be expressed as that number raised to the power of 1/3. So, the cube root of x⁴ is equivalent to x⁴ raised to the power of 1/3. In general, when we raise a power to a power, we multiply the exponents. According to the rules for integer powers, (x⁴)¹³ = x⁴*¹³ = x⁴/3.
To simplify further, we recognize that 4/3 can be divided into 1 with a remainder of 1/3. The whole number part (1) represents x to the power of 1, which is just x. The fractional part (1/3) is the cube root of x, which cannot be simplified further without knowing the value of x.
Therefore, cube root of x⁴ = x (x¹⁹³) = x (x¹³), since we can't simplify the cube root of x without additional information about x. Thus, the simplest expression for the cube root of x⁴ is x¹³, which corresponds to the option c) x⁴¹³.