Exercise 64

The exponent of the resulting power after reducing the expression \( 2^{2^2} \cdot 2^{ 2^{ 2^2 } } \cdot \left( 2^3 \right)^{3^3} \) to its simplest form is:

  1. 117

  2. 101

  3. 113

  4. 107

  5. 105

Try to solve it before checking the answer.

  1. 101

According power rules, we have:

\[ \begin{aligned} 2^{2^2} \cdot 2^{ 2^{ 2^2 } } \cdot \left( 2^3 \right)^{3^3} &= 2^4 \cdot 2^{2^4} \cdot \left( 2^3\right)^{27} \\[2em] &= 2^4 \cdot 2^{16} \cdot 2^{81} \\[2em] &= 2^{ 4 + 16 + 81 } = 2^{101} \end{aligned} \]

Hence, the resulting exponent is 101.

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