Let's solve the given mathematical expression step-by-step:
[tex]\[ 27^{-3^{-2^0}}+\left(\frac{1}{3}\right)^{4^{2^{-1}}} \][/tex]
1. Simplify the exponents step by step:
- Calculate [tex]\(2^0\)[/tex]:
[tex]\[ 2^0 = 1 \][/tex]
- Substitute this back into the expression for [tex]\(3^{-2^0}\)[/tex]:
[tex]\[ 3^{-2^0} = 3^1 = 3 \][/tex]
- Calculate the exponent for the first term:
[tex]\[ -3^1 = -3 \][/tex]
- Calculate [tex]\(27^{-3}\)[/tex]:
[tex]\[ 27^{-3} = \left(\frac{1}{27}\right)^3 = \left(\frac{1}{27}\right) \cdot \left(\frac{1}{27}\right) \cdot \left(\frac{1}{27}\right) \approx 5.080526342529086 \times 10^{-5} \][/tex]
2. Simplify the second term:
- Calculate [tex]\(2^{-1}\)[/tex]:
[tex]\[ 2^{-1} = \frac{1}{2} \][/tex]
- Substitute this back into the expression for [tex]\(4^{2^{-1}}\)[/tex]:
[tex]\[ 4^{2^{-1}} = 4^{1/2} = \sqrt{4} = 2 \][/tex]
- Calculate:
[tex]\[ \left(\frac{1}{3}\right)^2 = \frac{1}{3^2} = \frac{1}{9} \approx 0.1111111111111111 \][/tex]
3. Add the simplified results:
[tex]\[ 5.080526342529086 \times 10^{-5} + 0.1111111111111111 \][/tex]
4. Sum up the final values:
[tex]\[ 0.00005080526342529086 + 0.1111111111111111 \approx 0.1111619163745364 \][/tex]
Therefore, the result of the given expression is approximately:
[tex]\[ 27^{-3^{-2^0}}+\left(\frac{1}{3}\right)^{4^{2^{-1}}} \approx 0.1111619163745364 \][/tex]
