Evaluate. $\newline$ $\left(\frac{1}{81}\right)^{\frac{1}{2}} \cdot\left(\frac{1}{81}\right)^{-\frac{3}{4}}=$

Full solution

Q. Evaluate.

\newline

\left(\frac{1}{81}\right)^{\frac{1}{2}} \cdot\left(\frac{1}{81}\right)^{-\frac{3}{4}}=

Problem Understanding: Understand the problem. $\newline$ We need to evaluate the expression $((1)/(81))^((1)/(2))*((1)/(81))^(-(3)/(4))$ . This involves working with exponents and understanding the properties of exponents.
Exponent Properties: Apply the properties of exponents. $\newline$ When multiplying expressions with the same base, we add the exponents. So, we can rewrite the expression as $\left(\frac{1}{81}\right)^{\left(\frac{1}{2} - \frac{3}{4}\right)}$ .
Exponent Subtraction: Subtract the exponents. $\newline$ $(\frac{1}{2}) - (\frac{3}{4}) = (\frac{2}{4}) - (\frac{3}{4}) = -(\frac{1}{4})$ . $\newline$ So, the expression now is $((\frac{1}{81})^{-(\frac{1}{4})})$ .
Base Evaluation: Evaluate the base. $\newline$ The base is $\frac{1}{81}$ , which is the same as $81^{-1}$ .
Negative Exponent Rule: Apply the negative exponent rule. $\newline$ A negative exponent means we take the reciprocal of the base. So, $(\frac{1}{81})^{-\frac{1}{4}}$ is the same as $(81^{1})^{\frac{1}{4}}$ .
Fourth Root Evaluation: Evaluate the fourth root of $81$ . The fourth root of $81$ is $3$ because $3^4 = 81$ . So, the final answer is $3$ .