Sure! Let's solve the expression [tex]\( M = 216^{9^{-3^{-0^7}}} + 3(149)^{16^{-3^5}} \)[/tex] step-by-step:
1. Simplify the Exponent:
Start with the innermost part of the exponents.
- Zero Power Calculation:
[tex]\[
0^7 = 0
\][/tex]
Thus, we have:
[tex]\[
-3^{-0^7} = -3^0 = 1
\][/tex]
Then,
[tex]\[
9^{-3^{-0^7}} = 9^{-1} = \frac{1}{9} \approx 0.111111
\][/tex]
2. First Term Calculation:
[tex]\[
216^{9^{-3^{-0^7}}} = 216^{0.111111}
\][/tex]
Approximating [tex]\( 216^{0.111111} \)[/tex], we get:
[tex]\[
216^{0.111111} \approx 1.8171205928321397
\][/tex]
3. Simplify the Exponent for the Second Term:
- Calculate the power:
[tex]\[
3^5 = 243
\][/tex]
Thus:
[tex]\[
16^{-3^5} = 16^{-243}
\][/tex]
Given [tex]\( 16^{-243} \)[/tex] extremely small, it approximates to almost [tex]\( 16^0 \)[/tex]:
Thus:
[tex]\[
16^{-243} \approx 0
\][/tex]
Then,
[tex]\[
(149)^{16^{-3^5}} = 149^0 = 1
\][/tex]
Therefore,
[tex]\[
3 \cdot 1 = 3
\][/tex]
4. Final Calculation of M:
Adding the two terms together:
[tex]\[
M = 216^{0.111111} + 3 = 1.8171205928321397 + 3 \approx 4.81712059283214
\][/tex]
Thus, the value of [tex]\( M \)[/tex] is approximately:
[tex]\[
M \approx 4.81712059283214
\][/tex]
