When dealing with exponents being raised by another exponent, we just multiply the powers and keep the base the same.

\(\displaystyle (5^{-8})^{-12}=5^{-8*-12}=5^{96}\)

Evaluate the exponential term inside the bracket first.  Use the product rule for exponents to simplify.

\(\displaystyle (2^2)^3 = (2^2)(2^2)(2^2) = 2^{2\cdot 3} = 2^6 = 64\)

The term inside the bracket becomes:

\(\displaystyle [1+64]^2\)

Simplify the term inside.

\(\displaystyle 65^2 = 4225\)

The answer is:  \(\displaystyle 4225\)

When the quantity of the terms of a base raised to a power is also raised to a power, we can use the product rule for exponents to expand this expression.

\(\displaystyle (X^Y)^Z = X^{YZ}\)

Multiply the powers together.

\(\displaystyle (4^{150})^{5} = 4^{150\cdot 5} = 4^{750}\)

The answer is:  \(\displaystyle 4^{750}\)

\(\displaystyle When\ raising\ a\ power\ to\ a\ power\ multiply\ the\ exponents.\)

\(\displaystyle (8x^3)^2=8^2x^(^3^*^2^)=64x^6\)

To simplify this expression, since the powers are outside of a quantity of a power, we can multiply the powers together according to the power rule.

\(\displaystyle (x^a)^b = x^{ab}\)

Simplify the expression.

\(\displaystyle (2^3)^{-2}+(4^0)^3 = 2^{-6}+4^0\)

Change the negative exponent into a fraction and simplify.

\(\displaystyle 2^{-6}+4^0 = \frac{1}{2^6}+1 = \frac{1}{64}+\frac{64}{64}\)

The answer is:  \(\displaystyle \frac{65}{64}\)

The fraction inside the parentheses can be rewritten as a negative exponent.

\(\displaystyle (\frac{1}{3^3})^{-2}=(3^{-3})^{-2}\)

Using the power property of exponents, multiply both exponents together.

\(\displaystyle (3^{-3})^{-2} = 3^6\)

Simplify this value.

\(\displaystyle 3^6 = 3\cdot3\cdot3\cdot3\cdot3\cdot3 = 729\)

The answer is:  \(\displaystyle 729\)

Determine the inner term by using the additive rule of exponents.  When the bases of a certain power are similar, the powers can be added.

\(\displaystyle (3^{11}\cdot 3^{10})^3 =(3^{11+10})^3 = (3^{21})^3\)

Use the power rule to simplify this term.

\(\displaystyle (3^{21})^3 = 3^{21\times 3} = 3^{63}\)

The answer is:  \(\displaystyle 3^{63}\)

According to the rule of exponents, we can distribute the power of two by multiplying the powers.

\(\displaystyle (2x^{55})^{2} = (2x^{55})(2x^{55}) = (2^{1\times 2}x^{55 \times 2})\)

Simplify the terms.

The answer is:  \(\displaystyle 4x^{110}\)

According to the property of the power rule for exponents, 

\(\displaystyle (x^a)^b = x^{a\times b}\)

The exponents may be multiplied if the exponent is outside of the parentheses.

\(\displaystyle (2a^{120})^{5} = 2^5 \cdot a^{120\times 5}\)

The answer is:  \(\displaystyle 32a^{600}\)

In order to simplify this, we will need to distribute the power of 20 across both powers inside the inner quantity.

\(\displaystyle (a^2b^{25})^{20} = (a^{2\cdot 20}b^{25\cdot 20})\)

Multiply the powers.

The answer is:  \(\displaystyle a^{40}b^{500}\)