To simplify the expression \( \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \), we can start by recognizing that multiplying a fraction by itself can be expressed with exponents.
The given expression consists of four identical terms, \( \frac{1}{3} \), multiplied together. This can be rewrittens as:
\[ \left( \frac{1}{3} \right)^4 \]
Next, we express \( \frac{1}{3} \) as \( 3^{-1} \):
\[ \left( \frac{1}{3} \right)^4 = (3^{-1})^4 \]
Using the power of a power property \( (a^m)^n = a^{mn} \):
\[ (3^{-1})^4 = 3^{-4} \]
To express this with positive exponents, we can write:
\[ 3^{-4} = \frac{1}{3^4} \]
Now, calculating \( 3^4 \):
\[ 3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81 \]
Thus, we have:
\[ 3^{-4} = \frac{1}{81} \]
As the final answer in exponential form with positive exponents only:
\[ 3^{-4} \]
Hence, the final simplified expression is:
\[ 3^{-4} \]
1 answer