Final answer:
The equivalent expressions in the question are option (b) 1/(x^-1) and option (d) x^(1/3) • x^(1/3) • x^(1/3), since both simplify to x. The other options are not equivalent.
Step-by-step explanation:
The question asks which of the following expressions are equivalent and requires a justification for the reasoning.
First, let's examine option a. The expression 4√x3 signifies four times the cube root of x, which is 4x1/3.
Next, option b is 1/(x-1), which is x raised to the negative one power. Following the rule that a negative exponent denotes a reciprocal, we can rewrite this expression as x1. This simplifies to just x.
Now let's look at option c. Here, we have 10√x5 ⋅ x4 ⋅ x2, which is the product of the 10th root of x raised to the power of five and x raised to the power of four and two respectively. We use the rule that when we multiply like bases, we add the exponents, so this simplifies to x10/10+4+2 or x11.
Lastly, in option d, x1/3 ⋅ x1/3 ⋅ x1/3 shows the same base x being multiplied by itself 3 times, with each time raised to the 1/3 power. Adding the exponents, we get x to the (1/3 + 1/3 + 1/3) power, or x1, which simplifies to just x.
In summary, the expressions 1/(x-1) and x1/3 ⋅ x1/3 ⋅ x1/3 are equivalent because they both simplify to x. The other expressions, however, do not simplify to x and are therefore not equivalent to each other or to the two that do.