%5E(4%2F3).jpeg "(-27)^(4/3)")
Understanding (-27)^(4/3)
This expression involves a negative base and a fractional exponent. Let's break it down step by step:
Fractional Exponents
A fractional exponent like 4/3 represents both a root and a power.
- The denominator (3) indicates the type of root we're taking (in this case, the cube root).
- The numerator (4) indicates the power we're raising the result to.
Solving the Expression
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Cube root: Find the cube root of -27, which is -3 (since -3 x -3 x -3 = -27).
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Raising to the power: Raise the result (-3) to the power of 4: (-3)^4 = 81.
Therefore, (-27)^(4/3) = 81.
Key Points to Remember
- Even roots of negative numbers are not real numbers. However, odd roots of negative numbers are real numbers. This is why we can find the cube root of -27.
- Fractional exponents can be interpreted as roots and powers. This allows us to simplify expressions involving them.
By understanding these concepts, you can effectively solve expressions involving negative bases and fractional exponents.