%5E-1%2F2.jpeg "(9/4)^-1/2")
Understanding (9/4)^-1/2
The expression (9/4)^-1/2 might seem intimidating at first, but it's actually quite simple to solve using the rules of exponents. Let's break down the process step by step:
Understanding Negative Exponents
A negative exponent indicates the reciprocal of the base raised to the positive version of the exponent. In other words:
- x^-n = 1/x^n
So, in our case:
- (9/4)^-1/2 = 1/(9/4)^1/2
Understanding Fractional Exponents
A fractional exponent represents a root. The denominator of the fraction indicates the type of root, while the numerator represents the power to which the base is raised.
- x^(m/n) = (n√x)^m
Applying this to our expression:
- 1/(9/4)^1/2 = 1/(√(9/4))^1
Solving the Expression
- Simplify the square root: √(9/4) = 3/2
- Raise the result to the power of 1: (3/2)^1 = 3/2
- Take the reciprocal: 1/(3/2) = 2/3
Therefore, (9/4)^-1/2 = 2/3.
Key Takeaways
- Negative exponents: Indicate the reciprocal of the base raised to the positive version of the exponent.
- Fractional exponents: Represent roots where the denominator indicates the type of root and the numerator indicates the power.
- Understanding these rules allows you to simplify even complex expressions like (9/4)^-1/2.