%5E3%2F4.jpeg "(-81)^3/4")
Understanding (-81)^(3/4)
The expression (-81)^(3/4) involves fractional exponents, which can seem a bit tricky at first. Let's break it down step by step.
Fractional Exponents: A Quick Recap
A fractional exponent like 3/4 represents both a root and a power. The denominator (4 in this case) indicates the type of root, while the numerator (3) represents the power to which the result is raised.
In other words:
- x^(m/n) = (n√x)^m
Where:
- x is the base
- m is the power
- n is the root
Solving (-81)^(3/4)
Applying the concept from the previous section, we can rewrite (-81)^(3/4) as:
(-81)^(3/4) = (⁴√-81)³
Let's solve it step-by-step:
- Find the fourth root of -81: The fourth root of -81 is -3, as (-3) * (-3) * (-3) * (-3) = 81.
- Cube the result: (-3)³ = -27.
Therefore, (-81)^(3/4) = -27.
Key Points to Remember
- Even roots of negative numbers are not defined in the real number system. This is why we focus on the fourth root of 81, which is 3, and then incorporate the negative sign to account for the negative base.
- Fractional exponents can be simplified using the root and power relationship. This helps break down complex expressions into manageable steps.
Understanding these concepts allows you to confidently tackle similar expressions involving fractional exponents.