value of 81 to the power of – 1/2
The correct answer and explanation is:
The value of 81−1281^{-\frac{1}{2}} is 19\frac{1}{9}.
Explanation:
To solve this, let’s break it down into smaller steps.
The expression 81−1281^{-\frac{1}{2}} can be understood as a two-part operation:
- Negative exponent: A negative exponent means you take the reciprocal of the base raised to the positive version of the exponent. In other words, a−n=1ana^{-n} = \frac{1}{a^n}. So, 81−12=1811281^{-\frac{1}{2}} = \frac{1}{81^{\frac{1}{2}}}.
- Fractional exponent: The exponent 12\frac{1}{2} represents the square root of the base. For any number aa, a12a^{\frac{1}{2}} is equivalent to a\sqrt{a}. Therefore, 8112=8181^{\frac{1}{2}} = \sqrt{81}, and since the square root of 81 is 9, we have: 8112=981^{\frac{1}{2}} = 9.
Now, putting it all together: 81−12=18112=1981^{-\frac{1}{2}} = \frac{1}{81^{\frac{1}{2}}} = \frac{1}{9}
So, the value of 81−1281^{-\frac{1}{2}} is 19\frac{1}{9}. This shows the relationship between negative exponents and square roots.