Final answer:
By understanding properties of exponents and roots, it is shown that both 1/x^{-1} and √x³ simplify to x, proving they are equal.
Step-by-step explanation:
To prove that 1/x^{-1} is equal to √x³, we need to understand the properties of exponents and roots in algebra. As per the properties of exponents, given a non-zero base x, x to the power of -1, written as x^{-1}, is equivalent to the reciprocal of x, or 1/x. Conversely, a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.
Similarly, when we consider the cube root of x³, represented as √x³, it signifies the number which when multiplied by itself three times gives x³. The cube root of x³ thus simplifies to x, since (x) * (x) * (x) = x³. The processes of cubing and taking the cube root are inverse operations.
Therefore, from the properties mentioned above, it follows:
Since after simplification both 1/x^{-1} and √x³ equal x, we can conclude that 1/x^{-1} and √x³ are in fact equal. This demonstrates that using the rules of exponents and roots in algebra, we can transform and equate different expressions.
In conclusion, the understanding and application of exponents and roots are essential in algebra to simplify and prove the equality of different algebraic expressions.