2%2F3.jpeg "(−64)2/3")
Understanding (−64)2/3
The expression (−64)2/3 represents a fractional exponent. Understanding how to solve this problem involves recognizing the key concepts of exponents and roots. Here's a breakdown:
Exponents and Roots
- Exponent: An exponent indicates how many times a base number is multiplied by itself. For example, 2³ means 2 * 2 * 2 = 8.
- Root: A root is the inverse operation of an exponent. The nth root of a number is the value that, when multiplied by itself n times, equals the original number. For example, the cube root of 8 (∛8) is 2, because 2 * 2 * 2 = 8.
Fractional Exponents
A fractional exponent like 2/3 combines both exponent and root concepts. The numerator of the fraction indicates the power, and the denominator indicates the root. Therefore, (−64)2/3 means:
- The cube root of (−64) raised to the power of 2.
Solving the Problem
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Cube root of (-64): The cube root of -64 is -4, because (-4) * (-4) * (-4) = -64.
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Squaring the result: (-4)² = 16
Therefore, (−64)2/3 = 16.
Key Points to Remember
- Negative bases: When dealing with fractional exponents and negative bases, remember that an odd root of a negative number is negative, while an even root of a negative number is undefined.
- Calculator use: Most calculators have functions for calculating fractional exponents.
- Fractional exponents are often used in advanced mathematical concepts like calculus and algebra.
By understanding the basics of exponents and roots, you can confidently tackle fractional exponents like (−64)2/3 and many other mathematical expressions.