Which expression is equivalent to x1/3
The Correct Answer and Explanation is :
The expression ( x^{1/3} ) is equivalent to the cube root of ( x ), denoted as ( \sqrt[3]{x} ). This relationship arises from the properties of exponents and roots.
Explanation
- Understanding Exponents: In mathematics, exponents indicate repeated multiplication. For instance, ( x^2 ) means ( x \times x ), and ( x^3 ) means ( x \times x \times x ). The exponent can also be a fraction, which corresponds to a root. The expression ( x^{1/n} ) signifies the nth root of ( x ).
- Cube Roots: The cube root of a number is a value that, when multiplied by itself twice (i.e., three times in total), gives the original number. Mathematically, if ( y = \sqrt[3]{x} ), then ( y^3 = x ). Hence, if we rewrite ( x^{1/3} ) in root form, we express it as ( \sqrt[3]{x} ).
- Why Use Fractional Exponents?: The use of fractional exponents can simplify complex calculations and expressions. For example, when simplifying expressions involving powers and roots, having a consistent notation with exponents allows for easier manipulation of equations.
- Properties of Roots and Exponents: Both forms, ( x^{1/3} ) and ( \sqrt[3]{x} ), are interchangeable, which can be beneficial in various mathematical contexts. For instance, when solving equations or simplifying algebraic expressions, being able to switch between these forms can make the problem-solving process more straightforward.
- Examples in Practice: If you were to calculate ( x^{1/3} ) for ( x = 8 ), both ( x^{1/3} ) and ( \sqrt[3]{x} ) yield the same result: ( 2 ), since ( 2^3 = 8 ). This shows that understanding both forms is crucial for grasping mathematical concepts related to powers and roots.
In conclusion, ( x^{1/3} ) is equivalent to ( \sqrt[3]{x} ) due to the definitions and properties of exponents and roots. Recognizing this equivalence is fundamental in algebra and helps streamline calculations involving cube roots and fractional exponents.