Answer :
To evaluate the function [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex] for [tex]\( x = 3 \)[/tex], we will substitute 3 into the equation for [tex]\( x \)[/tex] and then simplify the expression.
Here are the steps:
1. Substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex]:
[tex]\[ f(3) = \left( \frac{1}{3} \right)^3 \][/tex]
2. Simplify the exponentiation:
[tex]\[ \left( \frac{1}{3} \right)^3 = \frac{1^3}{3^3} \][/tex]
3. Calculate the numerator and the denominator separately:
[tex]\[ 1^3 = 1 \quad \text{and} \quad 3^3 = 27 \][/tex]
4. Place the results back into the fraction:
[tex]\[ \left( \frac{1}{3} \right)^3 = \frac{1}{27} \][/tex]
Our final answer, as a reduced fraction, is:
[tex]\[ \boxed{\frac{1}{27}} \][/tex]
Here are the steps:
1. Substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex]:
[tex]\[ f(3) = \left( \frac{1}{3} \right)^3 \][/tex]
2. Simplify the exponentiation:
[tex]\[ \left( \frac{1}{3} \right)^3 = \frac{1^3}{3^3} \][/tex]
3. Calculate the numerator and the denominator separately:
[tex]\[ 1^3 = 1 \quad \text{and} \quad 3^3 = 27 \][/tex]
4. Place the results back into the fraction:
[tex]\[ \left( \frac{1}{3} \right)^3 = \frac{1}{27} \][/tex]
Our final answer, as a reduced fraction, is:
[tex]\[ \boxed{\frac{1}{27}} \][/tex]
